Mint does not have a deep root system and I doubt if a deep container will have much effect, if any, in the plant's growth, but I would not transplant into a permanent container less than 6 inches deep. Width would be of more concern as mint spreads wildly.
Mint plants generally prefer a wide and shallow container. This is because mint has a shallow root system and grows quickly, spreading out horizontally rather than growing deep. By planting mint in a wide and shallow container, you can give the roots plenty of room to spread out and grow, while also ensuring that the soil does not become too compacted.
In addition to choosing the right container, be sure to use well-draining soil and water the plant regularly to keep the soil moist but not waterlogged. Mint also prefers partial shade to full sun, so be sure to place the container in a location where it will receive some shade during the hottest part of the day. With the right care, your mint plant should thrive in its new container and continue to provide you with fresh leaves for culinary use.