Bike tires are different than cars tires. Its basically impossible to hydroplane a slick road bike tire due to the elliptical contact patch. As long as the tire has air, you shouldn't have handling problems.

Getting back to your original problem. Put that tire on your rear wheel. Reinforce the inside with a boot or a bit of duct tape if you'd like. Ride it until it wears out. Periodically inspect it, if you see a significant bulge there, that indicates the tire is failing and should be replaced. Once it wears out, put a new tire on the front, and move the front tire to the back.