Do the LHR transfer. LHR is actually easy, you'll just have to do immigrations for transfers (follow the purple signs). There are lots of lounge options at LHR depending on the amount of connection time you have as well.
EDIT: Forgot to add that LAX will have the new Polaris lounge open Jan 12 2019, not sure when you're traveling but keep the LAX Polaris lounge option in mind.