Well, I could not comment on the original thread I mentioned, because it was in a Christian-Only section of this forum. It seems there are Christians who are very well able to do it.
Considering that it was explicitly Christians who gave this advice, I think this cute little attacks on Muslims in inappropriate.
Interesting though that you mention it. What you said is wrong: Muslims are, by their religion, also not allowed to lie "if it forwards their purposes", and definitly not encouraged. This, basically is a lie, often repeated by anti-islam sources.
Muslims are, though, allowed to lie, if telling the truth would place them in unnecessary danger or hardship.
Quite relevant for this situation. Some Christians have repeatedly claimed that doing business with "sinners" (like baking wedding cakes for same-sex-couples, doing photos, etc.) would "go against their religion" or "make them celebrate the sin".
You might consider this "unnecessary harship". A Muslim in this case might be religiously justified to resort to false excuses.