You *might* be able to checkin for the LX flight yourself to at least get your own seat selection on the day of travel: I've managed to do this with other LX codeshare in the past, albeit for LHR-ZRH on an AC flight number - AC didn't print me a BP for this flight, they printed a "get your BP at the airport" non-BP.

To do this you'll need to use the advanced checkin, with the LX flight number that is operating the flight (NOT the codeshare), and using ZRH-BOM as your airports:
https://checkin.swiss.com/ck.fly#ck_retrieve

Note: I wasn't able to get this to work on a Brussels airlines ticket/codeshare (Brussels flight number, LX operated), I have no idea what happens for UA codeshares (in this case Brussels airlines offered limited seat selection on the LX flight at time of checkin - limited in that they only showed the second half of the economy seats for selection).