I'm not sure you can get much help from the type of commands found in the passage... one observation I think can be made about the OT commands is that often (especially in lists of very disparate commands) there may be very little relationship between any two such commands, and there are frequently commands that seem very out of place with those immediately before and after.
Keep in mind that the reason given for not doing either of these things (men or women) is that they are "an abomination to the LORD"... that means that how it is perceived or observed by others is not part of the equation. To me, that point argues less for "warfare" context and more for a "holiness" requirement.
Again, I have no way of knowing or even guessing what a man might do with a menstrual cloth... it might have been something we have simply no way to even imagine today... although it may have been as simple as using it as a belt.
Remember that during this time, clothing was so valuable a commodity that it was used as collateral for loans (Exo. 22:26-27), fulfillment of debts (like Samson's bet), and the soldiers at Jesus' crucifixion cast lots to see who would get to take home a blood-soaked piece of cloth (something today that would be considered bio-waist). People's attitudes towards cloth and clothing were very, very different during bible times... and very foreign to us today.
I looked a little more closely at the Hebrew here. Here is a "word for word" translation (and granted sometimes they don't make a lot of sense - but here it is):
Woman not fulfill of armaments (to prevail above) mighty men (warriors); neither shall a warrior be clothed in women's (his wife's) garments (to take on shape of women): for all who do such are detestable to the Lord your God.
I used "The Blue Letter Bible"; looking both at words and the roots they come from for this interpretation.
So in essence yes - do not use female combatants as human shields and don't dress male combatants as women to avoid attack from the enemy. In other words; if your going to fight wars, fight them like men! Using cowardice tactics is morally abhorrent. Even today, using unarmed civilians to cover military targets is against the Geneva Convention for the rules of warfare.
As per your "menses cloth" interpretation - here is another example where that interpretation would present issues: eunuchs.
Eunuchs:
I don't know how much you know about eunuchs and how men were made eunuchs. Eunuchs were usually employed to protect the king's family and were selected among men who were of honorable service to the king. Sometimes they'd been soldiers.
The traditional custom of Babylon and Africa was to select an honorable man who was mature enough, (yet still young enough to serve as a guard) who often times had already had his own family. Becoming a eunuch was usually a voluntary process. And being a eunuch could mean either castration, or having your entire external genitalia removed.
Now the records of how this was done; we have more from ancient Chinese sources than records from the near east. And usually the "surgery" was performed with a sharp sword after the individual had been somewhat sedated. Usually the shock of the action of the "surgery" knocked the poor soul unconscious.
First order of business was to prevent him from bleeding to death. There are some rather big arteries that supply blood to the penis. So they would pack the area with bandages using direct pressure until the blood clotted. The "hope" was that using a sharp enough sword would make a clean cut that causes the arteries to retract into the body and naturally "shut off", to prevent massive loss of blood. It's the same idea with amputating a limb. If you slice evenly through the flesh fast enough to cause the arteries to retract into the body; the person has a fairly good chance of survival.
Then they'd use some sort of antiseptic to prevent infection, along with inserting a "plug", which was usually made of sliver to prevent urine from coming out. If the poor soul survived the next 24 hours; his chances of surviving improved significantly. If he survived 48 hours; his chances of living were pretty good. The overall survival rate of such surgery was not very high (at least according to 19th century Chinese records); but the ancient world did know a little more about sanitation than the Victorian era did.
The silver plug implement was made specifically for and was now the property of the eunuch. He would need it for the rest of his life to prevent him from urinating on himself. Of which he'd have to wear some sort of "garment" to keep it in place.
Now the more radical form of creating a eunuch was not particularly uncommon for those who were personal guards of the king's family. The reason for such a radical procedure, was that castration alone does not prevent the ability to have sex. And this is why this was not done to just anyone; nor for that purpose of guarding royalty, was it done involuntarily.
To do that to someone involuntarily would only make them more bitter as well as dangerous. So this was not something that would have been commonly done to war captives or manual laborers just for the sake of doing it.
Also, in ancient China; (I don't know if this was a practice in the near east too) the eunuch got to keep his body parts. They were usually stored in a special jar and it was a huge shame to the eunuch if he'd lost his jar; or if someone had stolen it. Stealing or destroying a eunuch's jar was a serious offense in ancient China; punishable by death.
Other than beliefs of being restored in the afterlife - I'm not sure what the significance of the jar was; but it was a type of "commodity", as eunuchs who had their jars were more valuable than ones who didn't. To be a eunuch with a jar was a status symbol that you'd served the king and this wasn't something someone did to you just to be vindictive.
Clothing:
Now you gave the example of clothing being used for collateral and also the soldiers dividing Jesus's clothes. Some of Jesus's clothing was probably still useable (like sandals) and the cloak probably was not bloody.
Jesus would have had a cloak, sandals, a head piece / prayer shawl, a tunic, and a sleeveless undergarment that went to the knees which was called a "kethoneth". It was cold that time of year, so Jesus may have also had at least one other "shirt". (Which were garments layered between the tunic and the kethoneth for warmth. His clothing for that time of year, was probably 100% wool.)
It was not uncommon for soldiers to keep clothing of captives of a certain status. Most people who were crucified did not possess clothing that was of any value. Common criminals who (may have went from flogging) to crucifixion site usually did so naked, seeing how they were crucified naked. Jesus was "redressed" at the very end to stand before the crowd. He'd been "presented" as the soldiers had "dressed Him" (in a horse blanket and briar crown) to the leaders of the nation prior to this and when Pilate could not get anywhere with them; he appealed to the crowd.
Now being flogged before crucifixion is stated to have been "the custom"; yet there are records of crucifixions where the crucified was alive for days. So, "was everyone flogged to hasten their death"; is of legitimate debate historically speaking. (I don't know the answer to this.) According to military records; (which are a bit sparse because military discipline was dealt with differently) it was usually one or the other, not both. If someone was sentenced to death by flogging, they used different implements for said task than what was to be used for "chastisement" alone.
A "chastisement" was not to be fatal and if the "executioner" killed someone they were not suppose to; then they forfeited their own life. The Roman army was very strict about its discipline and that is one example of how. Crucifixion was a public execution for a reason. Where as "flogging until death" was not necessarily public.
Certain people could not be executed via crucifixion; including Roman citizens, females and soldiers. Citizens who were executed were beheaded (considered an honorable death). Suicide was another option (retained burial rights). This was often the choice taken by soldiers who'd served honorably. (I.E. a jailor who's prisoners escaped due to no fault of his own.) Soldiers who showed cowardice were executed by those in their command; usually by being beaten to death; i.e. "running the gauntlet" as was later called in history.
Pilate's idea was to "chastise" Jesus and let Him go. He hoped this would appease the rulers; they'd "feel sorry for" Jesus and not demand that He'd be crucified. Jesus was most likely "chastised" according to Jewish law / tradition (39 stripes) which was of common jurisprudence in Roman law too. This would have been done with a leather implement that did not contain alterations to inflict more damage.
Theologically speaking, for Jesus to endure a just punishment to atone for sin, laws for flogging from Deuteronomy 25:3 would have been followed. Jewish custom became 39 stripes just in case the flogger miscounted. Also, since Rome was the civil authority that governed Judea and you are to obey the civil authority; Jesus had to be executed by them.
Jesus was not flogged as had been depicted in Mel Gibson's "Passion of the Christ". Jesus never would have made it to the crucifixion site had He been whipped like that. Jesus walked from the Praetorium (site of trial) which was in the Antonia Fortress, across the court of the gentiles of the temple complex, through the triple gate, across the red heifer bridge to the Mt. of Olives, which was were He was crucified.
So having been someone of notable reputation. This is why the soldiers divided His clothing and cast lots for His cloak.