Interline Luggage
The BA agent was not incompetent, you were simply asking the wrong carrier. This is about LA interlining to BA not the other way around. The thread above suggests that LA will not and that is consistent with OW policy which is not to require it. You will need to check with LA as to whether anything has changed. Unless you can obtain an affirmative commitment from LA ---- highly unlikely ---- you must plan on checking bags to LHR and then rechecking them at LHR.

Immigration

TIMATIC suggests that either an airside or landside transit is possible for a Russian national traveling on a Russian passport though LHR. Prudence suggests having the e-ticket receipt printed out and available for the LA check-in agent. This would permit the Russian national to recheck his bags at LHR no matter how the transit is configured.

Transiting without a visa is possible for:Nationals of Russian Fed. holding confirmed onward tickets making an airside transit at London Gatwick (LGW), London Heathrow (LHR) or Manchester (MAN) on the same calendar day to a third country (excluding Ireland (Rep.)). Passengers holding confirmed onward tickets passing through United Kingdom immigration to make a landside transit to a third country on a flight that departs before 23:59 the next day.