Also LAX based and have done LX and LH in J multiple times in the past 7 months. Here's what I will say (I'll only reference hard product, as service really depends on the crew):

In J:
Hands down LX wins - while the cabin looks older/dated than the 747-8i (LH LAX-FRA), the seat is much better. The private seats (the best, IMO, are the odd seats on the "K" side -7K is my personal favorite) are great. In J on LH, I try to get a "middle" seat downstairs so nobody climbs over me, and I don't have to climb over anybody else. However, I still have bumped knees with my seat mate, or they still out in the aisle a bit - but that might have something to do with how I sleep.

However, if you're UA GS, you presumably have some GPUs to use. This is why I've moved to LH because I'm near 100% for upgrades from J>F. I know it's not a fair comparison at all, but since it's the reality since I sit in F. For that reason I'd book the LH through FRA (assuming F looks reasonably open).

However, LHR wasn't my final destination, so as others have mentioned, I'd look into direct options.