The F-150 can do this, but by charging the batteries you don't have.

It could also probably run the house off the 240V outlet in the bed, but only if you knife switched the solar + grid off and just ran the house off the truck.

When the sun came out, you could knife-switch the house back, and charge the truck.

Alternatively, you could rip out your solar and put in the one brand that Ford works with (SunRun, I think), and then the truck would be able to bi-directional charge with the house batteries using the solar, or something like that.

(Yes, this is all dumb, especially since there are standards that should allow a dongle to be made to make this work transparently, and to just automatically do the right thing".)