Should it be:
or
It should be “grammar,” first of all.
It should be “Grammar check”, and if you’re referring to some product in the store, it should be “No, we have no rat’s asses” if the product name is “rat’s ass”. If the product name is “rats’ asses” then it should be “No, we have no rats’ asses”.
If there is no product, and no name, then either way is grammatically correct, IMO.
I believe it should be rats’ asses, since you’re referring to multiple asses from more than one rat.
Correct. To get more than one rat ass, you need more than one rat.
I couldn’t give a rat’s ass.
If I had a dozen rats’ asses, I might spare one.
Unless we’re supposed to presuppose a possibility of a rat with more than one ass, I’d think it needs to be “no rats’ asses.”
Dr. Alphonse Mephisto would like a word with you. 
Yes - that’s what I was thinking. Thanks.