Welcome to our community and thank you for your question.
If I were you I'd just pay the fine. Usually there is no way to avoid it.
The only way to reduce it is to show them afterwards that you in fact had a valid ticket (for example if you have a monthly ticket that you forgot at home). That way you can prove that you didn't dodge the fare. In this case you will just have to pay a few Euros as an administrative fee. But if you didn't buy a ticket at all this won't work of course.
In general BAMF only checks whether you have a right to be granted asylum based on your interview and the situation in your home country. If you get in trouble with law it's subordinate to them in the asylum procedure. However, I found this document according to which criminal acts might have implications later on. For example it can happen that someone doesn't get a residence permit. But I don't think that's going to happen just because you've been caught without a ticket once. (http://www.fluechtlingsrat-bayern.de/tl_files/PDF-Dokumente/Beratungsmaterialien/Haubner%20Strafverfahren%20Auswirkungen%20auf%20Asyl%20u%20Aufenth.pdf
Just pay the fee and everything should turn out well. Fingers crossed.