OP will not stay airside, he has to change in ORDeal to Terminal E from the UA terminal via the train.
At departure time of LX 009 E terminal is a Zoo, expect long wait at the always understaffed security. It seems to me that the 6:00 pm wave of departing flights comes as as a surprise for the TSA everyday.

The US authorities don't allow LX to depart from the "domestic" terminal, don't kow why ,as LH does it.
The LX F lounge is or will be refurbished for 1 Mio $, in the meantime LX F will us the SAS lounge which is quite nice.

There are no UA worn carpet clubs in Terminal E.

IMHO taking the CLT-MUC flight on LH and connecting via Limo transfer to the 30 min MUC-ZRH is the far better alternative. This is what I always do, especially in Winter.