We had a similar situation a few years ago at the WL. We were initially booked with 6 people in a deluxe club level room, 5 adults, one child, and one under 3 so we all fit no problem according to Disney. Then we simply decided we didn't want to all have to share the space the entire week, so we ended up adding 2 additional rooms and split up. Me, my DH, and 2 sons kept the Deluxe CL room and the others had regular rooms. So now we were the only ones who had CL access. We were *hoping* that they might look at the situation - that they actually got the cost of 2 additional rooms out of us for the same amount of people, and that they were all at least initially listed on our deluxe room as guests and we would all fit under the occupancy of the CL room- and let the other 3 have CL access, but they would not. We hadn't expected it, but had been hoping and thinking it would be nice of them. We also didn't take it any further than simply asking the CM at the concierge desk, since while it would have been good customer service of them to look at the big picture and see it was a bit of an unusual situation they certainly had no obligation to do it. We did then have to have them remove the additional names from our room to get the additional adult fees removed, because we were still paying extra for more than 2 adults in the room (we had been willing to leave them on the room and pay the extra adult fee and then let them have the CL access). I still wonder if we had gotten the right manager and explained the entire situation if they might have let us, but it wasn't that big of a deal. Sure, they could have paid more depending on availability and gotten CL rooms on that floor, so we figured we weren't going to make a big deal of it. I also wonder if we had just left their names on our room and paid the extra adult fees for them if their bands would have just let them into the lounge. We didn't try that prior to the discussion. The 3 adults were apparently allowed to be listed in 2 rooms, because they were for the first night before we had them removed from our room. Nothing was mentioned or flagged for any of us when we all checked in.
Now having said all of that, your story is different given the medical situation. Not technically different, since it's the exact same situation regarding room occupancy and rooms booked, but I would like to hope they would look at your big picture and grant access. In your case I would likely take it to a manager if the first CM i asked said it wasn't allowed. I tend to think they will allow it if they take the time to understand the entire situation, and don't just knee jerk a "no" because the one room isn't CL, but that's just my opinion.