Deut. 22:5 says nothing about military attire, crossdressing or modesty.
The issue is "mixing things"
Mixing things, both in
Deut 22 and in the parallel passage in Leviticus chapter 19, are the central issue.
Do not mate different kinds of animals.
“‘Do not plant your field with two kinds of seed.
“‘Do not wear clothing woven of two kinds of material.
Lev. 19:19
Do not plant two kinds of seed in your vineyard; if you do, not only the crops you plant but also the fruit of the vineyard will be defiled.
10 Do not plow with an ox and a donkey yoked together.
11 Do not wear clothes of wool and linen woven together. Debt. 22:9
It is in this context of not mixing things we find
Deut. 22:5. It is another example of things we do not mix.