Weatherproof junction/extension box with 8AWG wiring - selection and mounting problem

The problem I see is your wire fill with 4 ea #8 wires counted at 3 cubic inches each you are at 12 without a clamp. I would probably be looking at a double gang bell box. One thing that regularly has to be done in industrial facilities is modifications to boxes or gutters. I would verify your local AHJ will allow this on residential. I would cut the back of a bell box out to slide over the existing device box and mount it directly to the wall this will look better and provide the additional space to splice the wires. Since the existing box is proud of the wall a bead of calking across the top or even around that box prior to sliding it over the existing box would make it waterproof. I have done this with both boxes and gutters coming out of concrete walls and steel plate. Your stucco would be the same as a concrete wall.

Trying to splice heavy wires in two tight a space is how wires get the insulation damaged and end up shorting out on the box or they vibrate through at a later date.

Honestly, if it were me, I'd make full use of the proudness to solve the rough surface problem. I'd get a 2-gang outdoor rated extension box - what makes an extension box special is it has an large opening at the back that is 1-gang sized, that basically makes it a box extension not another box. I say "choose the box" so you know how big its footprint is.

Then, for that footprint + maybe 1/4" margin... I'd either mill away the stucco or use epoxy filler and shims to bring the surface true flush with the existing metal box. That's all "body work". My go-to would be West System epoxy, since it lets you "dial-a-bondo" to the thickness you desire, and also use an adhesive filler (or none at all) for gluing or sealing. Also the very slow cure time is nice for epoxy novices. And mix it with no filler and you get a paintable sealant.

Then mount the extension box using the 2 original threaded holes in the metal box, caulk it on, and let the caulk cure.

From there, conduit system of your choice onward to the new subpanel, using THWN wire as Ed Beal discusses. Remember you complete assembly of the conduit before you pull the wires, so no plumbing elbows lol. Or if conduit isn't your cuppa, run cable out a weatherproof fitting - but remember you need #6 cable not #8.

Note that your #8 wire cannot carry 60A. I see no markings on the wire, making it either NM-B or UF-B type cable. Both of those are confined to 40A ampacity. The breaker must be changed to a 40A breaker. If that's not enough for your loads, you will need to run a new #6 cable. Sorry.