My breaker box is built into the wall. I can put my existing door on the outside, but that doesn't seem ideal as it leaves a gap. Is there a cover that will fit inside this? If so, what is it called? Tried many searches, but either they don't exist or I'm not using the correct term.
You can not have an open space but you can modify the dead front (the cover that screws to the front that seals and holds the breakers in place. These covers normally are wider to hide the edge of the box and the edge of the Sheetrock. The cover can be cut down to the size of the box. With the cover / dead face on the box is closed and meets code. At this point to cover the hole a picture can be framed and put on a hinge or Velcro. In doing similar the only issue inspectors have had is where is the main disconnect? One owner blew up a photo of a new panel and that worked for that inspector another inspector wanted a label above the frame stating main disconnect inside. I have done this where the inspector did not comment on the “cabinet” door covering the panel. If your main disconnect is outside you can cover it but it needs to be accessible.
Holy smokes, no! Panel covers are not junction box covers!
The panel cover MUST be flush with the panel. There is no choice nor option to that.
It doesn't work at all to have the cover be out of position. The cover being flush is essential to the breakers not falling out!! That would be clear if you understood how breakers attach.
Use the surface mount version of the cover for that panel.
Which manufacturers usually are willing to sell separately.
The surface-mount version, instead of having a big flange to cover up the drywall edge, simply is bent 90 degrees at the edge. It adds maybe a 1/16" margin to the box size. You will still need to plane or saw down the 2x4's a little bit, but it should be feasible.
Honestly, I'd cut back the drywall and 2x4 a lot more.
The reason is, as usual, no thought was given to finish treatments for the inside edge of this frame. So I'd actually remove about 1-1/4" from all four sides, so there's room to frame it in stainable wood 1-by's. And I'd create that box of 1-x's on the bench, built so the inside is 1/4" larger than the panel cover.... (and this would work with the original panel cover too).
Then having cut back the 2x4 material and drywall, I'd reinstall the panel cover, then shim the frame/lining so that it is dead nuts square on the panel cover with 1/8" margin on every side. Screw it down. Then stain it with a tasteful stain. Finally I'd fit a cosmetic fringe around the opening to hide the ugly gap between box and drywall.
For bonus points, you could simply put a tasteful cabinet cover over the whole shebang, then you wouldn't need all that trim wood.