Can a level 8 player join a Large mushroom battle?
No. A Large mushroom needs a squad of 15, but a level 8 player can only field 10 Pikmin. You'll be able to join once you reach level 10.
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 β no matter how strong your Pikmin are.