United appears to fly BUR-SFO, albeit in a cruddy regional jet -- did you check them?
Can't help you with the BART station, however, it is what it is.
If you fly to SJC and you don't want to rent a car or taxi/shuttle (expensive), then you take the free VTA shuttle bus to the Santa Clara Caltrain station and then take Caltrain to San Fran. This will take longer than your flight to SJC.