I already know it means "Why don't you try swapping school with me?", "Why don't you, and I, try swapping school?", but I wonder if it can also mean "Why don't you try swapping school for me?", like "Why don't you try staying with me instead of going to school?".
Would this be right?
You + with me + for my school + your school + don't try swapping?
If so, then the first sentence can have two meanings, can't it?
As you can see, you have to use と after あたし to show who does the action together with あなた and you have to use と after あたしの学校 to show the thing that will be swapped. The things being swapped are あたしの学校 and あなたの学校 (あなたの学校 is swapped for あたしの学校).
I was wondering if あたし in the top sentence can be, figuratively, the thing being swapped, instead of who あなた does the action with.