I think you might find that to remove the 4th screw (which I feel is absolutely necessary to replace the valve) you will have to remove the black tank.
That's because in order to be able to get the broken screw out, you will need to use a "reverse tap". The reverse tap is inserted into a hole you drill in what's left of the old screw, and it then turns counter-clockwise, the reverse direction to the original screw.
I'm thinking that from my description, you can imagine that you will probably have to drop the belly skin, or at least a good part of it. I'm not sure how cutting other plumbing off will help you get at the broken screw, but of course, I can't see your installation.
The whole thing is probably something that could be done in a day, but if you want to do it yourself, that may not be the case. Learning on the job isn't always the fastest way to get things done.
I hope you have an alternate 'loo to use while yours is out of service.
If it's to be, it's up to me.