Shutting that off abruptly will cause a pressure pulse to travel around the house pipework, which could trigger water-hammer effects in the toilet cistern.
The toilet fill valve shutting off abruptly can cause water hammer which results in a continuing vibration due to ripples on the water surface in the cistern. I had this happen in a previous house and cured it by attaching a submerged small bottle of water to the ball-valve float arm to provide...
In that case you need a plumber. It's most likely just the cartridge which needs servicing or replacing (if you can source a matching one), but a plumber would probably want to replace the whole tap!
That diaphragm looks similar to commonly available ones. You might be lucky in finding a third-party one which fits. Failing that, I'd be making some sort of alternate support for a readily-available type of side-entry fill valve.
For aesthetic reasons a perfect-fit disk is appropriate, but if aesthetics aren't a concern then your approximate-fit disk would be fine when glued in.
The foam washer should seal ok. Even if the seal is not 100%, the water leakage would be a lot less than the water your dog shakes off trying to dry itself. :giggle:
In a similar vein, it would be helpful if there were a national official registry for wills. As it stands, executors may have to check with solicitors, banks and other organisations to track down a will when someone dies.