In terms of minimal equipment costs, one solution would be to rearrange the TV brands such that rooms each have two different manufacturer's TV's. Of course, this implies that you would be physically swapping a bunch of TV's. And, if more than half of your TV's are from the same manufacturer, you'll run out of unique pairs of TV's.

Stepping back a bit, I think that you will have a management issue for the remotes. In each room the staff will need to be aware of which remote matches the TV's. Perhaps one could tag each remote with a room number and set number. ("105-A")

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One arrangement that I have seen is that each bed has a wired controller that can operate the bed, call the nursing staff, and control the TV. You may have such a system in place or might be contemplating installing such as system.

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A novel approach would be to partner with a local college or tech school and have them brew up a bunch of IR translators. These would be small, inexpensive computer boards that could be placed behind each TV. The computers would receive arbitrary IR commands and translate them to control the specific TV that they are mounted behind. This is the scale of project that the school could handle and the students would love the challenge -- and, this could become a point on their resume.

In the rooms, there would only be 'A' and 'B' remotes and all rooms would be the same. The computers would be programmed to match their TV and, since the computers would be mounted to their TV, the only reason to reprogram the computer would be if the TV manufacturer changed. As TV's break, the TV and computer could quickly be swapped with another pair. (making sure that 'A' is swapped with another 'A')

"Reprogramming" would be straight forward because you probably have a limited number of TV brands. If this is the case, a set of "jumpers" on the computer board would select the appropriate manufacturer and 'A' or 'B'.