I have cut many old records and I've mounted a small clock assembly in them. Cutting one record by itself is very tricky. I always stack cut 2 or 3 to give it more support. I put blue painters tape on both sides and mounted the pattern on the blue tape. The tape peels off very easily and doesn't leave any marks. You have to cut at a very slow speed because the vinyl starts melting back together at higher speeds. On my Dewalt saw I used between 1 and 2, and a small blade. The biggest problem is all the fuzzies left. I used needle files, small sanding sticks, and even an old toothbrush. But after a while I got pretty good at it.
As far as what record to use. I Googled images of the original record label and printed the image on a blank CD label, and then I put the CD label over the record label. The CD label is just about 1/8" inches or so bigger than the record label, so it works out just perfect. So it looks like it's a Beatles album but it's actually an old Screamin' Jay Hawkins LP, or something like that. (nothing against Screamin' Jay Hawkins).
I made many different clocks and sold them for $40 at a craft mall in Wisconsin Dells, a huge resort area about 90 miles away. Sorry my picture's not the greatest, but you get the idea.
Ray F