I think if the face frame is sturdy, and sturdily attached with something that will reference it to flat (ie biscuits or dowels etc) on the face of the carcasse then it will likely pull it straight. If you cut a bit of mdf the same size and thickness as the carcasse uprights, the middle will flex as it doesnât have the structural ridgidity to not do. So if we remove the issue of not square corners (which having a very squarely cut back panel, and then flexing the carcasse to fit it when attaching it will fix the problem at the back edge) then the front will still belly in the centre of the run.
The face frame will fix this.
Other option is to build out of thicker material, or something sturdier like birch ply..
doweling the corners of the carcasse may help with the squareness of the corners, but wonât fix the bellied carcasse sides.
In your standard kitchen this is mitigated by jamming lots of units next to each other and fixing them off to each other (pulling them up)
With regards the back panel, clearance holes through the mud back panel, and holes exactly the diameter of the shank of the screw (not the size of the thread) will work fine, you could use pan head screws if you donât want to countersink.
I would personally start with the back panel, measure it as accurately as you can corner to corner, then itâs width at the top and at the bottom. If thatâs all fine, does it allign perfectly everywhere with the carcasse ? If not, take it off, and when attaching it again flex the carcasse ( or use clamps as spreaders) to allign it as well as you can, then the back of the carcasse will be pulled square. Start in one corner, then work along one edge, then the next edge.
Hope this helps.