The problem is you IGNORED that Jesus was also 100% man and man CAN be tempted and to be honest I doubt either of us know how much "God" Jesus had operating in him or as you equate it.... "supressed".
Personally this internal vs external temptation idea is meaningless as it only equates that you are immune to being tempted and you already have decided Jesus is immune to temptation which would mean he isn't a man at all. I say Jesus resisted temptation and lusting/desiring etc but did have to deal with it as equated in Hebrews 4:15
For we do not have a high priest who is unable to empathize with our weaknesses, but we have one who has been tempted in every way, just as we are—yet he did .
In this scripture someone who is immune to temptation entirely could not empathize with our weaknesses because essentially he was immune to it and that (to me) means he felt nothing of being tempted and no struggle to overcome it. Someone who has never suffered cannot emphasize with the suffering of another IMO.
Suppressed, or whatever Jesus I believe had to concentrate on being God and I also theorize that the like believers after pentacost the Holy Spirit showed up. It is quite possible that for the miracles he did as a child he called upon the Holy Spirit to do such and not of his own divinity the Bible doesn't really equate one or the other. Now when Jesus began his ministry the Holy Spirit was there falling upon him with his baptism by John.
Jesus through the power of the Holy Spirit indwelling him still not using his own divine powers could accomplish as a man anything that we could with the same power of God and the Holy Spirit acting under the Father's guidance would easily do what is needed in all situations and thus be doing the Father's Will in it. There were times when Jesus didn't do any miracles at all. To be honest it would be confusing to really know for sure if it were Jesus divinity or power doing everything he did or the Holy Spirit's power as also God doing everything or a mixture of one or the other. If all that were done in his incarnation as a man was done by the Holy Spirit acting as his "agent" of God then he would be totally Man essentially devoid of God for practical purposes.
I believe I read somewhere that Jesus set aside his divinity, you use suppress which to me is not quite the same in definition as it could equate a constant action instead of just turning off and turning on when needed. With Angels and the Holy Spirit protecting him he could easily be just a man in life till God's power is needed and thus be 100% able to empathize.
Because Jesus did not have a nature of sin like all of mankind inheriting it through Adam's sin he was giving the opportunity to lust but could resist it without the use of his deity it was more along the line of a choice to sin or not but likely he still had to feel and see things that we do and decide or choose to not go beyond desire in his heart (and mind).
In the end if Jesus could truly empathize with us, he had to be tempted and have the chance to doubt etc otherwise his empathy would be a sham. The Bible equates he was tempted in the Desert for 40 days and if he cannot be tempted then he wasn't tempted at all but the Bible disagrees because part of the temptation in the desert was feeling hungry and thirsty which he did and it was not comfortable at all he could have chosen comfort and fall into temptation. Jesus had faith in God (the Father) and the writings of the Bible that he would not die of hunger and thirst during that 40 days. Personally I would probably have died if I had to go that long.
Again, as I already stated. There is a huge difference between external temptation and internal temptation. A man can be tempted externally by a phone telemarketer, and have no interest in the telemarketer's junk he is selling. So this would mean there would be no internal temptation by the person being solicited or called. Jesus was tempted
externally in every way as we are. This is how is able to sympathize with our weakness in Hebrews 4:15 because He knows firsthand what it is like to be tempted as a human. It does not mean Jesus actually considered even for a half a second to do wrong. Again, God cannot be tempted by evil. For if you truly believe Jesus is God, then you must grab hold of the truth in James 1:13 and believe it.
James 1:13
“Let no man say when he is tempted, I am tempted of God: for God cannot be tempted with evil, neither tempteth he any man:”
The Scriptures also say that Jesus is HOLY, UNDEFILED, and SEPARATE from sinners.
Hebrews 7:26
“For such an high priest became us,
who is holy, harmless, undefiled, separate from sinners, and made higher than the heavens;”
So where did this internal temptation come from within Jesus? If Jesus could be tempted by evil, according to the Bible He cannot be God because God cannot be tempted by evil. Jesus is also said to be separate from sinners and He is holy, and undefiled.
Jesus is called the LAMB of God who takes away the sin of the world.
John 1:29
“The next day John seeth Jesus coming unto him, and saith, Behold the Lamb of God, which taketh away the sin of the world.”
You are suggesting that there is some kind of spot or darkness within my Lord Jesus whereby He could potentially do wrong in that you suggest that He was actually internally tempted by sin.
1 Peter 1:18-19
18 “Forasmuch as ye know that ye were not redeemed with corruptible things,
as silver and gold, from your vain conversation
received by tradition from your fathers;
19 But with the precious blood of Christ,
as of a lamb without blemish and without spot:”
Jesus is our spotless Lamb.
There was no spot within Him to do any wrong.
Jesus' is the pure spotless and clean Lamb who was able to cleanse us from sin. Jesus is holy, undefiled, and separate from sinners. But you want me to believe that Jesus is not separate from sinners (Which is contrary to Hebrews 7:26).
Jesus at His core is the ETERNAL Logos or Word (i.e. God).
Remember, the Word is God (John 1:1), and the Word was made flesh (John 1:14). The core essence of Jesus is that He is God. Yes, He was also a human, but it was not possible for Him to sin or do wrong because He is God Almighty.