If you cannot remove the two cabinets, its a little tricky to suggest a suitable solution without a picture. However, a possible solution. Would be to cut your stuff for the frame and rebate it such that the width if the rebate allows it to sit on the plywood and come flush with the edge. The rebate not only acts as a self aligning jig but also allows you to tack the pieces onto the frame whilst the glue dries. You will need to mitre the corners accurately to ensure a good fit with clean joins. A little bit of messing about and time consuming but should give you a solution.
The rebates can be cut in a Spindle Moulder, Router Table, table saw or with a rebate plane. My preference would be the first option but any will do the job adequately. The thing you need to ensure that the rebate is accurate and square. Any raggy edges will not be seen.
If you are also edging shelves that are also build in, the same applies, just place the rebate so that the wide edge is uppermost to keep the lines inside the cabinet correct. If you look up to to the cabinets, reverse the orientation of the rebate so that it look consistent.
