I think you have a couple of options.
REMOVING THE OLD RIVET/DRIVE SCREW:
First, whether it is a drive screw or a rivet, put a center mark in the center of the head (use an automatic center punch...the thing you press into the piece, then it 'snaps' and punches a center mark. If you don't have one of those, then you can use a center punch and a hammer. I personally recommend making a dot with a super fine Sharpie, hitting the center of that with a ***** punch, then hitting that with a center punch) Very carefull drill down in with a drill bit the size of the shank (you won't know this, so start with a small bit and move up, until the head of the rivet spins off. You are NOT trying to drill out the whole rivet, just remove enough of the center of the rivet head to remove the head without damaging or enlarging the original hole). Then, take a pin punch of the appropriate size and punch the rest of the rivet/drive screw through the hole.
Here is a video of the technique at the EAA website:
http://www.eaavideo.org/video.aspx?v=1657896937
REINSTALLING THE TRIM:
Now, you have a couple of options. Either find new SS drive screws and make sure they fit (usually driven in with a hammer) (you may have to go up one size to get a tight fit). Or, drill the hole out to the next size up of stainless steel POP (blind) rivet. These really don't look too bad and might be your easiest option, if you have a POP rivet gun already. Certainly won't be noticed if you drill out all the trim pieces and re-POP-rivet all of them. If you prefer a bit of a custom look, and you have access to the back of the piece, you could pick some hardware you like to replace the rivets/drive screws (I particularly like the look of socket head cap screws...it would lend a very industrial look to the tool box).
OTHER THOUGHTS:
Is their really no access to the back, or do you have to take all the drawers out to get to the back? I would seriously consider buying stainless steel solid rivets for this job, but I already have all the solid rivet driving tools--that wouldn't be the inexpensive way to do just one tool box.
Kev.
