Your pistol and ammo has to be in its own locked (non-TSA locks) case, but said case can be inserted into other luggage.
Personally, I’d advise against calling the airlines or asking anyone at the counter. Rather, find their respective policies online and print them, keeping the link handy on your phone. It’s exceptionally common for individual workers at airline counters to not be familiar with these respective airline policies, TSA requirements, or FAA laws, which can cause issues upon checkin. Quite commonly, I’ve faced counter clerks which insisted TSA locks were required, but upon presenting their own policies and TSA policies to them, proper process prevails.