There is a 5 fold increase in power requirement between the level 1 and level 2 chargers. You seem to be getting along with the level 1 charger,even though its slow.
The risk of adding the level 2 load is much higher, particularly since we do not know what the existing load on the service is, nor the actual condition of that panel. As I said earlier, I am a bit concerned with 50 or 60 years of Bay Area fog may have done to it.
If you can't get an electrician to inspect it and ok the additional load, you can live with what you have, trade the electric for a hybrid vehicle, or find a different place to rent.
I assume you have a list of charging stations or a phone app for them.