Can a level 22 player join a Large mushroom battle?
Yes. A level 22 player can field up to 40 Pikmin, and a Large mushroom only needs a squad of 15 to start — so you can join.
Your maximum squad size grows with your level, and each mushroom size has a minimum squad you must field to start a battle. If your cap is below that minimum, you can't join — no matter how strong your Pikmin are.