I would also go to a transmission shop. It's hard to find really good ones. I'd consider calljng around first and ask if they have experience with Toyota automatics. A lot of trans rebuild shops have never touched one. The failure rates are really low that used ones are usually cheap because there's no demand and they are so infrequently rebuilt that many shops simply haven't ever had a chance to work on them. I know locally most will sub out toyota transmissions to some national specialty place.
I don't think 1k lbs over weight would phase it at all. It's only got a little v6 pushing it. The same transmission was used behind V8's for a lot of years with a lot more torque - including supercharged v8s. And it holds up just fine behind supercharged v8 torque. Could you abuse it to the point of failure? Probably, but it's really hard to do.
The idea of a shaft failure seems like a good thought, but my issue there is that it's a v6. It shouldn't have enough torque to come anywhere close to snapping a shaft in that transmission. If you have shop space and tools, I'd start by checking trans fluid level one more time. Then drop the transmission pan and just see what's inside. Maybe the oil pickup is clogged or the oil pickup tube somehow came off. Visually check all of the electronic connections. Maybe even pull them apart and put back together. And double check the cable linkage to the shifter to make absolutely sure you're not somehow disconnected or loose and it's stuck in neutral.
I would also double check and make sure that the transmission in park is locking up the rear tires firmly. It might be in neutral and somehow the transfer engagement clutch is just friction closed and not engaging the teeth. Basically put in park and try to push it with maybe 2 people and see what happens.
If you do need to replace - I'd personally prefer a low mile used over a rebuilt. I've had so many bad rebuilt engines and transmissions that if it isn't an OEM rebuild or someone very experienced like an Aisin specialty shop, I'd rather have a used one. I think the going rate is around $1k for the used takeout. For example: here's one with under 60k miles for under $1k.
Used 2016 TOYOTA 4 RUNNER TRANSMISSION - OEM 3200004010 - Benzeen Auto Parts No affiliation, but I order from them periodically and have had good experiences so far. The reason is that they always have a picture or set of photos of the actual vehicle it came out of so you can see the damage and make sure it isn't likely to damage the parts you want.
Good luck. Hope it's something easy and cheap!