I would not risk LH coach. At least on UA, he has e+. So, going off of the worst case scenario (GPU doesn't clear), I'd rank your options in this order, from best to worst:
1) LH PE to MUC (A380 business is nicer than A340 IMO, but the transition from wherever the hell LH decides to park at FRA to the B gates for your LHR flight can be a pain)
2) LH PE to FRA
3) UA E+ (this is a 787, right? The seat width sucks, but is still better than LH E-)
4) LH E-