And if your a Military family, you probably have moved all over the US or farther most of your life, but you don't see excuses being made in that case. That's what it is in almost ALL cases against requiring an ID...excuses.
There have even been YT vids of interviewers going around in poor and minority communities asking if they have an ID or DL, and if was too difficult to get one, and I think all the respondents already had one, and quite a few minorities thought it was racist, in and of itself, to suggest that they were too poor, uneducated, or lazy to get one. If price is the big hurdle, then maybe we can cover "State ID" (not driver's license fees), for poor people. This shouldn't cost a lot in tax money to give out. Most people use a DL as ID rather than a State issued ID card anyway. If you get a DL, then it shouldn't be free as I assume if you have a car to drive, you can pay the $20 to get your own DL.
So what if it takes a bit of time, we are talking about a vote that happens every 4 years. Local elections are a bit more often, but either way, you have plenty of time to get one if you want one, it's not like major elections are announced 2 weeks prior to them happening. They happen on normal, routine schedules that can be planned for well in advanced. Occasionally there might be an unscheduled one when some politician dies or is arrested/fired for corruption, or some other unforseen, but in 99% of the cases, there is plenty of time to obtain an ID and register to vote. A Birth Certificate is something you only need to get once, then you keep it, and can re-use it if you move or need to get other documents that require it, it's not a big burden at all.
In fact, I think it's almost better to require at least SOME planning ahead of time before voting. I don't trust that the type of person who registers to vote the day before the election is going to be the type that is fully educated in the people or measures on the ballot. Last minute registration like this reminds of "impulse buys" at a grocery store, which are something you didn't plan, and don't think through very well. This is dangerous when it comes to something as important as voting. Before anyone chimes in about some people needing to register last minute because of their age or something, then this could be solved separately. I'm not suggesting that if you turn 18 just mere days before an election you are out of luck, but if you've had 4 years since the last election, then you have a planning problem, not something else.
People of lower income are also less likely to move out of their districts, so while they may be renters and more likely to change addresses or such (don't know, but that is what is posited by OP), most lower income families and people don't make big moves, but more local, like hey the rent is cheaper over in this part of town, or the next town over, so in many of these cases, they are still within their district. Heck, I've moved 4 times before buying a house, and all of these moves stayed within the same district, so voter registration didn't change except for an address change.