It seems they have already figured out some way to authenticate and tell whether a refund request is valid or not. I think they already mentioned the way they are doing it for now here in the very thread, anyways.
By the way, I do not think there have actually started to refund fund to this day, they seem to continue to figure out the orders and filtering those which are valid from attempts to claim balances from people trying to take advantage of the situation. If any significant percentage of their gamblers had already started to receive their funds then they would have appeared here showing those proof of reception of fund or something like it, also Betnomi would be eager to start to show some proof of refunds, but it is not the case for now.
With what is currently happening to Betnomi right now, I think that it's going to be difficult for them to pay those people because they're also having an ongoing legal battle that might need some funding and we know that cases like this are expensive and takes a long time to settle either by a settlement or a judgment from the court. It's a good thing that they're starting to list people that are looking for a refund from their casino so they don't have to worry too much about the list as time progresses.
You are being kind of optimistic if you think they can recover up 70 percent of their previous reputation, by the way. When refunds are finished and most of the people is satisfied with them, I would be surprised if they still have a 45 percent. Though, obviously betnomis plan is to work from there and slowly gain everyone's trust again through the months and years ahead.
That's probably the only way to approach this kind of situation, being optimistic. Optimistic that they will be able to return their money as soon as possible. Regards to their reputation, that's going to be rebuilt eventually and it's going to be mended quickly if they completed the refunds. Maybe when they finally did the refund, that's up to them on how they will be able to build their reputation again so for now it is difficult to speculate if they will rebuild that reputation or not.