Assuming an extra $215 isn’t a huge hardship for you or your friend, I would fly AF.

LAX - CDG direct, enjoy Paris then train to Amsterdam /Brussels in whatever order you want. Eurostar from either Amsterdam or Brussels to London and then finally return LON - CDG - LAX by plane. I wouldn’t bother with Eurostar back to Paris as a flight connection in CDG will be more straightforward for your return to the US.

Given that you are not going to Germany, the uneccessary stops in FRA or MUC just waste time and create extra potential for IRROPS.

Your multi-City search should be:

1. LAX - CDG
2. LON - LAX

On any Airlines.