I'm a very simple kind of guy, I like my basses without all the bells and whistles, so for me, a boutique bass would be a waste. If I'm going to pay a boat load of money (assuming an unlimited budget as you said) then I'm going to go for the real thing. As me having an unlimited budget is unrealistic, you could also assume that I'd have the time to play a bunch of vintage basses until I found the one that suits me right.
I've already somewhat made this choice I guess, as I have two 78 fenders and a 78 gibson (total coincidence I swear). I got the Jazz first, it was hanging on the wall at a local music store. I'll admit I'm a total sucker for blocks/binding. I picked it up and knew then and there that was the bass. It really spoke to me. I've played other vintage basses that didn't do it for me, but this one was just right. I got the gibson G3 because I was looking for a grabber and this was the first thing that came up. It's ok and there's certainly nothing like it in production now (except maybe the EBMM Big Al). The 78 fretless precision was something I ran into at a pawn shop and saved up some money to get it, mostly because it matched my jazz, but also because it plays so beautifully. I'm sure it wouldn't be too hard to find a maple fretless P-bass from any boutique maker, but for the price this was a nice piece.
As an afterthought, I know that if I were to go boutique, it woud be vintage style, either a P or a PJ, jazz neck with blocks and binding, matching headstock, with as-close-to-vintage-fender pickups/electronics. Essentially I'd want a duck dunn with a maple board and possibly a bridge pickup.