OK, I think I've got my head around it!
That's always the problem with a hospital tank, is how to keep its filter bacteria alive. If you were to get some fish to do that job (and let's face it, any excuse is a good reason to get more fish!!) you have to move them every time you need to quarantine a fish. Once the sick fish is better, you then have to move it back to the big tank, and throughly clean the hospital tank, otherwise you risk infecting the hospital tank fish.
In your circumstances, what I'd be inclined to do is to keep the hospital tank empty and dry, with filter media in its filter. When you need to use it, take the media out of it, take media from your external to fill the hospital filter (but no more than a third of the external's media) and replace the dry media into the external. Fill the tank with water, pop in the sick fish, with fully mature media in the hospital tank filter. The replacement media in the external will regrow its bacteria quickly, so there won't be an issue with a minicycle - that's why I say to only take a max of a third of the media from it.
When the fish is better, pop it back in the main tank, and clean and dry the hospital tank, and place it in storage, leaving all filter media where it is.
I hope that's clear enough, cos I've just read it back to myself, and it's confused me!