If you can't lift the fittings, you will still have problems from the extra thickness of your new floor. Plus the risk of possible problems with the fiddlinesss of it all. If you do use solid wood, there can be the added fun that you cut a paper template (which fits perfectly) transfer it to the wood (and cut it exactly) only to find that you can't get the wood in place because it won't fold and there is no room to move it around in the available space!
Given that it's only a small room, the cost of good vinyl flooring would not be huge, and some of it is very good indeed.
Given that it's only a small room, the cost of good vinyl flooring would not be huge, and some of it is very good indeed.