“Rim” and “wheel” are interchangable except for split truck (or heavy equipment) wheels.
Anyway, there is a technique called “roda force balancing” that balances the wheel & tire by psinning them with simulated road force applied. It presses a spinning drum against the tread while it runs. These can detect internal defects, poor concentricity, and other problems that regular balancing machines cannot. I’d be inclined to try spinning the wheels on one of these machines first. The store where you bought the tires may even have one.