Do separate VLANs behind the different SSIDs provide protection?

I would guess that the VLAN separation should prevent it, but perhaps there are implementation errors on the VLAN implementation inside of individual brands of routers?

Inter-VLAN routing shouldn't be done at the wifi access point, packets would need to be tagged coming out of the wifi AP and switched upstream, unless I'm mistaken about this.

That should definitely help. You still have to double-check the IP routing tables between the VLANs, but most of the time, that should prevent attacks between SSIDs.