I've dealt with similar problems many times in rental properties.
In my opinion, the piece of rubber and hose clamp is not the way to go, for three reasons. 1. It will only temporarily fix the problem. 2. There is a chance it will not even temporarily fix the problem. 3. Either way, you'll have to cut out some wall to do a permanent fix, so it's better to do it now before any water damage becomes expensive to fix.
It sounds like you know exactly where the leak is. I would cut out a square piece of the sheetrock, so that will be easy/simple to replace, maybe even with the same piece, that is big enough for you to work comfortably in the wall. Cut out a few inches of the copper pipe and just use compression fittings to replace what you cut out with some of that bendable copper pipe stuff. The best way would be to use a solid piece and solder it, though.