Moose, I think I can visualize part of your "upside down sled" idea, but 5 times the length and width would make it a 15" x 10" sled.
The hardwood cleats idea makes no picture in my mind. How would I hold the box on the under side of the sled?
With such a big sled, how would I see to place my box inside the jig? As it is, I have to pivot the piece a little while putting it in the jig so as to gently plunge the bit into the bottom of the box. taking off 1/32" of material to smooth out the forstner bit grooves.
I was off with the 5x sled size. Mentally I was trying to get your hands away from that bit. Maybe sled isn't the right word. More like 'handle'. See my attached diagrams (very quickly done, so use your imagination).
Green is router table, and green cyl is the bit. Yellow workpc. I don't show your fence that limits travel. You just need to be aware of heights so my cleats don't hit your fence. Cleats are in blue. Attached permanently to the red sled/handle. The red cyls are just grab points (handles for the handle). You could set your workpc over the non-spinning bit, and fit the cleat/sled down over the top. Lay it on top. Press down. You control the piece, even though it's not attached to the sled. Now push the workpc around in whichever lateral motion you want (climb cut even).
This whole idea, and any other, IMO, must be only for skim cleanup. Like the last 1/16. Take very small bites. Visibility might be an issue, but in a way it's more psychological, no? You have the stops/fences. You have the piece firmly pressed to the table. You can't see the inside of the box anyway... But, you could use 1/4" plexi for the sled.
Re the skim cleanup, here's an example that may help. I make solid body guitars. I cut the body shape out of an 8/4 blank with a bandsaw, zipping around the curves using a 1/4" blade and staying mebbe 3/16" from my line. Then it gets a ride on the bench and spindle sanders, removing wood all the way to just kissing the pencil line. All that's left is probably 1/32". Because there's so little wood for the router, I can run the Whiteside 2" CL spiral at full height, running it around that body, in both directions. There's no hint of danger or fear. But if I were to leave a fat 1/16" on there instead? I might lose something, and not just the workpiece. I see your situation as similar, just inside-out, and much smaller, closer.
Hope this helps a bit.