Could I just tap into the two wires in the panel that go to the pump?
You should still have a way to ensure the breaker panel isn't wired to the inverter. Some kind of a DPDT transfer switch, plug and cord, etc.
But I don't know if your inverter can start the pump. Induction motors draw a huge surge current, about 5x their nameplate current.
I had also considered one of your other suggestions: use the generator for the AC inputs on the inverter. So maybe I'll try that first. I guess my other question is how to bring the inverter ac outputs wires into the panel since I only have a Line and Neutral. There's a ground post on the bottom of the case but I'm not sure if I can use that. (I'll attach a picture). Maybe I need a step down transformer(?)
The inverter may have a "neutral" that is tied to chassis, and it may just have two isolated hots.
It is even possible its chassis is a midpoint between the two hots, i.e. a neutral. Some cheap 120V inverters are actual +/- 60V.
Check voltage to chassis and see what you've got.
Do you have a transformer recommendation to run the 110 outlets in the house (from the box)?
Depending on whether inverter outputs are isolated or have a connection to battery or chassis, different transformer types might be considered.
An isolation transformer, 240V in, 120/240V out (or two isolated 120V secondaries) is one approach.
A non-isolating 120/240V transformer (or a 120V to 120V isolation transformer wired as such) would be half the mass for same VA rating (because only half the power goes through magnetics, the other half is direct wired.)
You need to determine what wattage loads you want to run. And if any are motors requiring surge capability.
I mostly shop on eBay, look for something with suitable voltage windings and VA ratings. Toroid transformers tend to be more efficient.