Your observation that failure only occurs after starting the engine is really interesting. I can think of 2 causes based on those symptoms: vibration and heat (heat increases with current, which is higher when the alternator is spinning).

You might try poking around with a stick with the key on but engine not running, so see if you can induce a failure. Definitely leave the lights on for 5 minutes or so, without starting the engine, to see if heat eventually causes one to turn off.

While your perception is completely-off or completely-on, I would compare both lamps at night (when both are on) to see if they are of equal brightness. If RH is perceptibly dimmer, this suggests a lossy connection somewhere on that side, which translates to a hot spot. If your vehicle has the resistance wire harness, it is suspect.

You can check voltage at the 2 fuses, when 1 lamp is off, to determine if the problem occurs before or after the fuse. The headlight fuses are usually under the hood; locations may be identified in your Owner's Manual.