I understand that "It's the love between you and I" is supposed to rhyme with "When we laugh so hard we cry."
Unfortunately, it's just wrong.
It should be "the love between you and me." "Me" is the object of the preposition "between." It should be in the objective case, as it is in "Correct grammar gushed out of me," or "My education was a metaphorical support beneath me." I mean, think of it: you wouldn't say "between we," would you?(Would you??)
No, you would say, "Our common passion for the English language pulsed between us
So, may I make a suggestion? In order to have both your rhyme and your grammatical street cred, you should make it:
"When we laugh so hard we pee,
It's the love between you and me!
See? Grammar is a wonderful thing.