My comment is in reference to installing your new face frame; since the cabinet will be painted, I would glue the face frame to the sides and install with four 2” finish nails on each side. The nails will be set, spackled and painted, so they will not be seen.
Do you have room to add a 1/4” panel of ply to the inside? If yes, this 1/4” panel would tie the two pieces together.
There's room but the surfaces aren't flush as the plywood is around 1/4" thinner than the solid wood case it's mating to. Also, I may need the joint top not be perfectly planar. This whole cabinet tis out of whack and the glue up will need to be such that everything 'sort of' fits together. It will need to be clamped to coerce it away from the 'square' fitting of pieces.
It needs clamping because this cabinet is a huge PITA ... everything is out of whack. It's like playing 3D chess. It would have four times easier to just have torn this out and built it all from scratch.
The right stile (that's to say the broom closet's stile) which the new right case side attaches to is not parallel to the face frame. The stile is rotated so presents itself at an angle out of plane. It's significantly out of parallel so that if I attached the right cabinet plywood extension side to it that case side will not be square to the face frame. But 'the buck stops here' ... I'm not renovating Monticello, so I'm leaving it as it is and it's where I draw the line. Because everything is cockeyed and out of square it means my glue up is going to be more complicated as I need to force pieces into place and have them stay there. Simply putting glue on and pressing pieces together than walking away will end badly.
The plywood cabinet side-to-stile is connection is biscuit joined but it's not going to fit flush against the stile ... it's rotated out a little so it's square to the face frame. Again, the buck stops here. [I'd considered cutting the case side edge at an angle to make up for the defect, but my biscuit jointer is a 25+ year old Porter Cable with the worst designed fence and difficult to adjust in the best of circumstances and it doesn't do angles so I'd need to also make a beveled fence for it .... etc. etc. ...then I'd have a case side badly out of square to the face frame and highly visible ... so this is the balance I've struck].
Anyway, I need to assemble this carefully, clamp it securely, and apply the right pressure in the right places.