Can "No thanks" be used in place of "No need to say thanks"?
Solution 1:
Nonsense indeed. They are quite different, if only because "no thanks" idiomatically just means "No, but thank you for offering."
Funnily enough though, "No thanks necessary/required!" is the same as "No need to say thanks."
Solution 2:
No thanks is used to decline an offer from somebody.
'Do you want tea?'
'No thanks.'
No need to say thanks is used when someone thanks you for doing something.
So the two are not interchangeable in the context you provided.