According to the order sheet, they come with a Shimano UN53. That's pretty decent quality, part of the LX range, when replacing an extra fiver will get you a UN73 which is XT. Your crank puller will be fine. Before you invest in a new BB take the cranks off, clean and grease the square tapers and replace, some people say you shouldn't grease them but it's worked for me in the past, just another possibility!
I'd be surprised if it was the BB causing your problems, they usually last 10,000s of miles without any trouble. The only time I've had them fail it's been a grinding noise rather than a click and you could feel it through the pedals. As you say it's something to eliminate, without spending that much if you've already got the tools.
A cautionary tale - We were talking about odd noises on a club ride, somebody had had a drivtrain noise for weeks, it stopped when his handlebars snapped!