People used to do this quite a bit back in the day, but it hasn't been cost effective for decades. That said, who cares if someone just wants to do it?
In my opinion, use or get an LU-563 hook saddle B1811563000. I never heard of using the existing LU-562 saddle or a Singer 111W. I'm not at all saying that it can't be done, but I've never seen it done that way. LU-563 machines are almost 30 years old or more, some parts have to be available. Take up lever, if you can get new genuine, I would do that if you are not sure. It's a real problem to change this take up up lever in and out if it does not fit or work properly. B19055630B0 is the part number for the take up lever. Same for the hook. You can get a made in Japan hook, if you want to go heavy thread like the LU-1508NH you can use B1830563BA0. You could have everything correct but if the hook is bad from the start, it could be trouble. Like needles, hooks are very rarely of poor quality from a good Japanese or Italian manufacture. You also are going to want the bobbin case opening lever, B1824563000.
This is how I would start this project, and I'm interested to see how it works out.