I've done it connecting through PHL (UA -> US) and had no checked bag problems. We were in Business Class.
At the prices you quote, you're looking at coach fares. So the question is, which airline has the best coach?
You'd do best to cross the Atlantic in UA's coach (and secure Economy+ through status or buy-up) but get the E+ at time of booking! Then connect on LH from, say, MUC.