To be fair, the Padres don't really have any "crosstown competition". They don't have NFL, NBA & NHL like most other cities have. They only have the MLS that could potentially compete with the Padres. I think that helps the Padres' attendance a lot.
Not to make excuses for Nats attendance, as price and record are key downward pressures, but the Nats are competing with a lot more local competition than the Padres are sporting wise. I'd like to go to more Nats game, but ultimately I have to account in my budget for Commanders games. I imagine lots of people on this board have the same challenge with Caps, Wizards.
Padres are the biggest show in town by far. The Padres have a great setup being the big show in a city that is pretty big but not so big it has demanded a second Big Four sports league. And LA being congested with franchise means it's unlikely San Diego will get another one in the near future.