I'd only consider keeping the Tundra since you already have it. They have well proven reliability, and I think it will work well... maybe with a little tweaking and careful selection. I guess I don't like trading vehicles as often as some people...
I'll just tell you my experience...
The Tundra frame is boxed under the engine, quite sturdy C channel under the cab back to the forward leaf mount, and then transitions to wimpy C channel. This rear part flexes quite a bit torsionally when crossed up, and vertically on bumps. I made a 3 point mount for mine, and it has far exceeded my expectations at dealing with the torsion... and also handles great IMO, even with no swaybars, once I changed to Ironman FCP shocks. I can exceed the speed limit easily in turns, it responds well to sudden maneuvers, and isn't scary in strong gusty winds, which are not unusual in NM. However... the vertical flex can be annoying if I hit a series of bumps that are spaced to cause resonance. This happened even when unloaded on some concrete roads, but it's worse now. I'm pretty sure the frame is the main culprit, but the camper itself is surely flexing as well. The frames on 1-tons will be stiffer... they are all fully boxed now (torsionally stiff) and they also don't have to pass crash testing, which I think is the main reason why 1/2 tons struggle with the rear frame, since it needs to absorb energy in a collision.
AT had a camper on a Tacoma initially. All the others I've seen were on 1 tons. The Tundra is closer to a 1 ton in size and weight than it is a Tacoma.
If you have the DC 6.5' model (mine is the DC 8'), the bed overhangs the front mount by more than a foot, and this is a weak point. For any bed (stock or custom) that area should be reinforced.
Brakes are fine. In a panic stop you will very likely be able to stop quicker than you could with a 1 ton, and IMO this is the only sort of extreme braking event we should be worried about, since we are not racing or stupid enough to drag brakes on long descents.
I'd pay special attention to the rear springs and shocks. Aftermarket is geared toward lifts mostly and light loads, which isn't ideal for camper hauling. Idaho Spring sells 4-leaf additions (each side) to the stock pack for ~$600 shipped. I don't have them yet; still just have airbags. The Ironman shocks have very high and digressive damping which is what I wanted. This is good for handling. Custom valved 2.5 shocks would be ideal for more money. Mostly I did not want to deal with overhauls on those.
But if you want to go with a 1 ton, the F350 with 7.3 would be a good choice I think. Although if that was the case I think I'd look for a bigger camper for a 2-person full-time setup.