How to finish Butt joint and corner bead within 5″?

I created pass through door between my kitchen and dining room. I am done screwing sheetrock to cover framing. I need to finish this job with drywall tapping and corner bead installation.

Now as you can see in picture, butt joint and edge of door (where vinyl corner bead will be installed) is just 5" away from each other. How many coats of mud should I apply and in which sequence ? Can my 3rd/4th coat for butt joint be applied during corner bead coat ? I am assuming that my corner bead coat will be wide around 6" anyway to blend in corner beads.