Drawing a weapon after making a claw attack doesn't require an action.
Both you and your DM are incorrect in thinking that switching weapons requires an action; there's no rule that says this. What does require an action is interacting with multiple objects on your turn. But while your claws are a weapon, they aren't an object, and don't need to be interacted with before you can draw your crossbow.
In more detail:
As part of your turn you can interact with one object "for free":
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.
If you need to interact with more than one object in a turn, there's the Use an Object action:
Use an Object
You normally interact with an object while doing something else, such
as when you draw a sword as part of an attack. When an object requires
your action for its use, you take the Use an Object action. This
action is also useful when you want to interact with more than one
object on your turn.
If you are holding one weapon and want to sheathe it and draw a different one, that's interacting with two objects, and thus you need to use your action to do it all in one turn.
But: your claws aren't an object -- see the discussion in this question about what is and isn't an object.
The Tabaxi Cat's Claws trait specifically states that when you attack with your claws, it's an "unarmed strike", which is understood to mean not involving a held weapon or other object. (In fact, it can be argued that in order to make an unarmed strike with your claws, you must not be holding any kind of weapon in the attacking hand, since the strike is defined as being "unarmed").
Since your claws aren't weapons, you don't need to interact with them to "put them away". If your object interaction for the turn is with your hand crossbow, and you can do that alongside making a claw attack.
Also
In your question, you made a claw attack on one turn, and then wanted to make the crossbow attack on the next turn. Even if claws counted as objects (or if you were using a held weapon instead of your claw), you could do this. You could make your attack on the first turn, then stow you claw/weapon as your object interaction for that turn. On your second turn, you could draw your crossbow and then fire.