Based on the title of the question, it sounds like you want this cover crop to successfully compete with grass. Given that, I would recommend something like a white clover. It is very dense, can easily compete with grass, and is pretty low-growing. It also very well tolerates being walked on (source, my front lawn is entirely clover and it holds up just as well as grass). It also fixes nitrogen in your soil, improving it for the other things you want to grow there later.
I really wouldn't recommend mint for a few reasons. First, it's not all that short. My mint plants are easily 12 inches tall, and they are not even getting close to the amount of sun they should. I would anticipate 18 or more inches. Second, the structure of the mint plant does not appear as though it would tolerate being walked on at all - I would expect it to break/snap and stay flat on the ground. Last, while it will establish and spread (it is invasive after all), it is not particularly dense so I wouldn't expect it to prevent grass from growing - I can imagine it co-existing with the grass rather than choking it out.