"I lost my ID, I need a new one", "Sure. Please show me your ID so I can ensure you are who you say you are and issue you a new one", "I can't do that, I lost it", "Then please wait until you receive the certified letter attesting to the fact you are who you say you are", "But I need to vote", "You'll need ID for that", "I know, but I lost it", "Then you need to get a new one".
I don't understand the problem. If you do not have an ID you do not get to vote. The reason why you do not have an ID is irrelevant; without an ID they cannot verify that you are casting your own vote. Without this requirement it would be possible for people to buy other people's "stempas" (the letter that you get in the mail that allows you to cast one vote). That would be very bad because it allows wealthy people to cast multiple votes. This is also why there are strict rules about only one person entering into a voting booth, and why you are not allowed to take a picture of your ballot.
How do you know that voter fraud rarely if ever happens if you don't ask for an ID? Why does requiring an ID disenfranchise people? If people are prevented from getting an ID then surely the issue is that people are prevented from getting an ID, which is necessary for lots of important actions, and not that you can't vote without an ID? Making it about voter suppression makes it seem like the only thing that matters is that these people vote for the right party, and not how not having an ID impacts their lives.
Repeat.