They can vote, but registering where you don't actually reside is a violation of many state laws.

Kindly point to the relevant state law which requires you to be a physical resident of the state, and thereby disallows deployed members of the military and overseas voters from voting in the state. There may be some, but I've not heard of one.

In any case, your complaint was about the lack of representation in Congress for overseas citizens, excepting by those who fraudulently declare residency in the state. The Uniformed and Overseas Citizens Absentee Voting Act says the state must allow overseas voters to vote in federal elections, and therefore do have representation, and without resorting to fraudulent means.