Of course you can make the requests. I would simply ask them to put it “no handicap accessible room”.
As you know - the chances of getting the HA room in the Tower Club are much higher than other places due to them only having 5 rooms total.
Couple of suggestions -
IF they end up giving you that HA room - and them not being able to accommodate a different room - ask to get a room in the Atrium Club but keep access to the Tower Lounge since it is more quaint and not as crazy busy.
Offer that you’d be willing (IF you are) to change rooms if they have availability to shift you around. I’ve done that one time when they only had Bay Lake and I really wanted MK view. We changed to MK view after two days.
I would highly suggest you do this upstairs with the Tower CM’s. If they make you check in downstairs just take what they give you and deal with what I mentioned upstairs. The Tower Crew has some of the most amazing CM’s in property and they will be accommodating any which way they are able to and they will accommodate. Just be honest and tell them why that room really doesn’t work for you family and what I suggested up there to show that you are willing to make it work on your end too. You might even end up with some pixie dust