[Arran]

I think that if you strip off the finish from the mahogany (or whatever lookalike species it is) veneer, and just dampen that, then some of that should work it's way to the structure underneath. You don't need to soak it you just need to get the moisture balance right between the inside and the outside of the panels, since the outside was sealed more then the inside, the inside absorbed the moisture from whatever damp basement it came out of. I think you are doing things the right way, although you will likely need more blocks and clamps, but you just have to be patient about it.
Just out of curiosity are the joints of the cabinet loose? Like could they be removed by undoing a few screws or whatever? Give it a week or two of the treatment, at worst you would have to make some replacement panels for it if it doesn't fix itself. Are the side panels made out of a plywood or are they made out of boards laminated with veneer on both faces?
If you are wondering whether it is worth the effort I think it probably is, if the production statistics are right the MX cabinets are not that common. Is it worth paying a pro to fix the cabinet I don't know because I usually do that sort of thing myself. Maybe you could farm out reproducing the side panels to someone else if you have to, but they would likely need the old ones to go by.
Regards
Arran