The Classic Player Jazzmaster I played at the store in Dublin (California, not Ireland) was a fantastic instrument and I'd love to own one. Regardless of the ToM bridge or any of the other alterations.
I would, however, replace the stock pickups with Novak or Lollar. The CP series has pickups that are wound hotter and use different magnets than the original Jazzmasters did, and Novak and Lollar are also a considerable deal more careful in their manufacturing. I bought a set of Tele pickups from Fender (via Allparts, who didn't give me a receipt in the box) and they came to me broken before I opened them.
If you get an original or vintage reproduction Jazzmaster, the bridge thing is something to note, but can be solved by replacing the original saddles with Mustang bridge saddles, which are available on the Internets. The problem also isn't anywhere near as bad as it is on the Jaguar, and having owned a Jag for roughly two years now, I'll tell you it ain't as bad as it's made out to be.
I will say that I fear the Jazzmaster and Jaguar are going the way of the Strat though. Fender is doing their absolute best to homogenize them both. Fortunately, it kills off a lot of the demand for vintage ones, so their prices are staying relatively low (as compared to vintage Teles or ES-335s).
Also guys last night I saw a guy playing a mid-60s ES-330 and I really had to resist the urge to break my glass over his head and make off with it. Dude can't play for shit anyway, I don't know why he has that thing at all.