the stack size in a PDA M can grow to hold at most k symbols. What kind of language is L(M)? Prove your claim.
I think the answer for this is this: The language L(M) is a regular language if it can hold at most k symbols. Since it can hold at most k symbols, the stack size is therefore finite. Furthermore, we can create a DFA that will accept L(M), and therefore we can conclude that this language L(M) is regular.
But I am having a debate with my friends whether this is correct or not
You are correct, such automata accept the regular languages.
One way to argue this is by showing how to construct an NFA for any of these limited-stack PDAs. The construction is easy, as you have suggested and has been suggested in the comments: