In the meantime, it vibrates from underneath, and feels like it's holding back. It's very frustrating, and embarrassing. Is this caused by the odd tire?
How different is the odd tire? It's a different make, but is it also a different nominal size from the other three? Or, is it the same tire size, but different tread depth, and if so, how different?
The vibration could be due to an odd-size tire, especially when accelerating in a lower gear because the AWD transfer clutch is more fully engaged under those conditions. But it also could be a failing propeller shaft U-joint, a broken rear transmission mount, or other causes.
There's good suggestions above, such as trying it with the FWD/AWD fuse inserted to disable the AWD, and changing the ATF.
There's too many possibilities, especially when we're not there and can't experience the symptoms first-hand. It's better to eliminate them one-by-one. Take it step-by-step and report back on the results.