I think that matching the texture will be the most challenging part of this...

I agree, but for "Get it as smooth as possible." I'd only knock down any bumps sticking out of the current thinset wall. Drywall mud will stick to it, and it will stick better to a rougher surface.

What did the piece of plaster look like? That is very clean lathe to have just removed original plaster from it... and there are two remaining screws that would more likely be used for drywall or plaster boards.

I think that it would be easier to buy a 4x8 sheet of 3/8" drywall and cut it to roughly fit in the hole. Screw it in and mud on top (more labor, harder to mess up), or shim it out so that you just mud the joints (requires a bit of skill, but isn't too difficult).

Edit: Mudding the joints involves joint tape. If you want to match the texture, peel back a few inches of the existing texture so that you can tape from the old to the new...

Have you done some radon testing lately? Might be a good idea so that you don't need to incorporate remediation later.