Then buy 5 new tires, OR put the spare tire onto the bent rim.

The tire store will tell you that a 10-year old tire is not serviceable and should be replaced. You certainly do not want one tire on the ground that's different than the three new ones, so you need at least 4 tires to match on the ground, and the different one in the trunk. If the tire in the trunk is 10 years old, then it will just keep getting older and therefore less and less serviceable.

All you need to do before you put my plan into action is to look to see if your spare rim is the same as the others on your car.