I agree with the Mexican approach. One option: bus or San Diego trolley to the Mexican border, walk across, Mexican bus (they're nice!) to a major city, get on a plane. Get a MX visa on the other side of the border crossing.

I don't know yet what their security for non-US flights is like. But when I flew from a mid-sized Mexican city (Mérida) to the US, there were no scanners in sight, just the walk-through metal detectors. At the gate, everyone's bags were moderately searched and everyone received an old-fashioned, barely-there patdown. The staff were polite, I wasn't sexually groped, and everyone got the same treatment.

I suspect the procedures at the gate were an accommodation to TSA rules. I'll soon find out what kind of security happens for flights from Mexico to London.