OK, now then . . . . fortunately you've been around and know what's involved, and there are great techs in Nashville.
I'm going to assume when you say 'set neck', it's a glue joint, right? So . . . . . IF there's not a lower profile bridge to be had, then the only thing I can see is one of four things:
a) Steam the neck off (as you would a box guitar) and reset it to work for your action, or . . . .
b) Counter-sink the appropriate bridge, ala Spector, so you can have proper saddles and adjustments, or . . . . .
c) Steam off the fingerboard and lower the surface where it meets the neck, or . . . . .
d) New Bass Time !
I'd vote for either a) or b), provided there's no big finish issues involved, and this is an axe you want to spend this extra money on. There's enough Gibsons and acoustics floating around NashVegas that I'm sure there are qualified 'steam' guys. Or on the other hand, the guy that built it may be best for a neck reset. Routing a small recess to countersink the bridge would be the least invasive. D) by far is the easiest cure.
Your dilemma is not unheard of: The late John Entwistle's basses had action so low, they were virtually unplayable by anyone except him, so you are in good company.
Best of Luck !