Thanks for all the advice!
The flight is in late August, and the full routing is actually BUD-MUC-LHR-EWR (I only mentioned the MUC to EWR part because I thought that was all that would be relevant given that UA doesn't fly out of BUD and can't budge with partner availability).
There is an available BUD-MUC on LH at ~6am that would get me into MUC in time for UA's 9:30am MUC-EWR, but at that point, I'd rather just leave the day before. I can't see the agent would be THAT nice.