I opened the seams as much as I could, scraped out as much of the old foam as I could, filled the cavity with polyurethane foam, and then clamped it all back together. This repair tactic was a great success. Mostly.
I put some cheapy kevlar/nylon strings on the bass, tuned it up, and plugged it into an amp but there was barely any sound. For those that are unaware - and that's probably most - the pick-ups are two coils positioned underneath flexible steel discs on which the bridge sits. You can adjust the clearance between the coils' pole-pieces and the discs. Anyway, I removed the pick-up assembly to find some of my foam had expanded into and around the pick-ups essentially dampening the vibration of the discs under the bridge. Got it all cleaned-up without issue and she sounds great now.
On a side note, there are a total of three discs stacked in each hole in the pick-up plate for each foot of the bridge. The signal I'm getting from the pick-ups is adequate, but is lacking in the lows, particularly the B string (obviously I have the bass strung with a low B). I wonder if I removed one of the discs, if that would make them more flexible and possible get me a little more energization of the coils and maybe bring the B string up a bit. Anyone have any opinions on this?