I'd be slightly more optimistic than most here - I've done QR to BA and it was about 45 minutes. However, that is still going to be a tight transfer and is dependent on an airside bus, which isn't all that frequent. I'd only consider it if your LAX flight departs after midnight CET and/or you're certain you'll have WiFi on the plane, and can then book a fare that allows free same-day change (NOT Avios, and not basic economy) so that you can do a same-day change with confidence (and presuming there is a viable option to change to!). Even then, there's an element of risk as FRA-LHR has been known to sell out.