Is my bike capable of swapping for longer cranks? I have a cross-check at 52 cm with a casual drop handlebar set up. It's a relatively square frame compared to other bikes I've ridden in the past. The cranks feel very small, like I'm not using my whole range of motion to pedal. I'd be curious to run longer cranks.

The problem is that because of the short wheel base, on turns it's possible for me to accidentally kick my tire when I adjust my clip position to a relatively comfortable position.

Is my frame simply too small to fit longer cranks?
Yes, your bike is capable of using longer cranks.
Longer cranks will likely cause more toe overlap, which is generally expected and widely tolerated on small-framed 700c bikes. Here is Sheldon Brown's opinion of toe overlap, from his online glossary:

On many bicycles, especially those with smaller frames and full-sized wheels, it is possible for the front fender or tire to bump into the rider's toe or to the toe clip. Some people worry a lot about this, but it is rarely a significant problem in practice.
The only time it can happen is when the handlebars are turned quite far to the side, as only happens at very low speeds.
Many, many people ride bicycles with fairly severe overlap with no practical problems, sometimes having to make a slight adjustment to their pedaling habits at very slow speeds.
On smaller-size bikes with full-sized wheels, it is usually impossible to eliminate overlap without causing adverse fit/handling issues.