How/when did Claudia figure out who Regina’s father is?

How?

Mundane means (DNA testing) are possible. Once Claudia got the time-ball from Alt Claudia, it would've been trivial to collect Regina's and all candidate fathers' DNA samples, then go to a time period in which paternity tests are available.

When?

I don't think when she found this out is particularly important. It took place during the years she spent investigating the true origin of the knot.

Why?

Which leaves us to the more important question of why she decided to DNA test Regina. We know she was uncertain who Regina's father was. She said to Tronte that for some time she hoped he were the father, which implies lack of certainty.

We also know that at some point Claudia realized some people were of the knot (exist only because time travel exists), but others like herself and Bernd weren't of the knot.

Before all of this, finding out Regina's true father would, at most, be a curiosity to Claudia. After all, paternity tests were already available in the 1980's but Claudia didn't bother, as it wasn't hugely important. However, once she realized some people were knot children, Regina's parentage became hugely important to her strategy. If Bernd's the father, then Regina is not of the knot, and destroying the knot is a viable tactic. If Tronte's the father, however, then Regina is of the knot, which means Claudia must seek to maybe change or alter the knot so Regina lives; destroying it (and thus deleting Regina) would be out of the question.