Can mobs spawn in water near Mushroom Islands?
Solution 1:
If you press F3 you should be able to see the biome you're in. If you go in the lake and see it's not Mushroom Field or Mushroom Field Shore, then mobs can spawn there, underground, on dark platforms etc.
Ideally you could use that space for a mob spawner, even if I'd suggest going over the ocean for that.
I would like to point out that because it's connected to a normal biome like a forest or an ocean, mobs can still spawn there and move in the biome. The only workaround is to build barriers on the seashore and add torches.