If you can fit a correct diameter spare I'd check craigslist for something used. Or even discount tire. It's probably worth the $50-100 to not worry much about it. Keep in mind that a new tire with tread vs a bald one will be different circumferences even in the same size (which is yet another good reason to always do 5 tire rotations).
That said I would absolutely run the wrong size tire to get off a trail. I would try to avoid engaging any lockers unless absolutely necessary in that case, and I'd keep the speeds low. You'll be generating additional heat in the center diff regardless of whether you're locked or not as the diff has to continuously slip.
@Madtiger, when it comes to donuts, it depends on the vehicle, but many RWD-only or FWD-only vehicles have spares which are both smaller in width as well as diameter.
That said I would absolutely run the wrong size tire to get off a trail. I would try to avoid engaging any lockers unless absolutely necessary in that case, and I'd keep the speeds low. You'll be generating additional heat in the center diff regardless of whether you're locked or not as the diff has to continuously slip.
@Madtiger, when it comes to donuts, it depends on the vehicle, but many RWD-only or FWD-only vehicles have spares which are both smaller in width as well as diameter.