How should I paint over a large patched area on a wall?
It's hard to completely eliminate signs of a patch. Try to get the patch as level with the rest of the wall for starters. You're still going to have some irregularities from the mud but if you feather it out it's less noticeable.

Just mud the outside of the patch for now. Do multiple light coats until you get out to a 12" taping knife. That should leave a gradual slope that will be less noticeable.

After the outside dries scrape down the inside edges a bit to get them a little even then  do two or 3 coats to fill in the inside of the patch so that the inside is flush with the inner rim from the outside mud.

Use a straightedge and a work light on the side of the wall to check for irregularities.

Once you're happy with it, prime then paint. Using a paint with very little sheen will help to hide any imperfections too.