Essxjay:

On the Frankfurt-Prague leg, I think you must be talking about the LH3306 or UA3736 flights or the LH3310 or UA3738, which are both LH flights with UA as the code-share.

The same applies to the Budapest-Frankfurt leg. So it looks like you are stuck with Lufthansa whatever you do.

But I would be happy about that if I were you.

Regards,
MF