Modifying size/limiting mule inventory

only way I could think of is to open a custom invetory when the player opens the mule inventory. that way you can manage the size and who is allowed to open it without a problem.

 

if you implement that ^^

 

If you're going to have max the amount of slots a mule has, just store items in the mules inventory.
Drops on death, too.

If you're going to have more than the max amount of a vanilla mule, make a new Inventory instance and save the items.
Get the entity ID or such, too, btw.
Id rather just use the first slot of it's vanilla inventory to keep a stick or something with a displayname of it's UUID