The biting will get better as he loses his baby teeth - the adult teeth are less sharp and won't draw blood as easily.
Make sure you are reacting in the right way when he mouths you:
http://www.jersey.net/~mountaindog/berner1/bitestop.htmTry to get your fiancee to try to play with toys with him, rather than with each other directly. I mean - it's fine for your dog to be rough with a toy, play tuggy, shake it about etc - but trying to play with your fiancee's clothing or limbs isn't fine. So try to control the games with toys.
I think he could be understimulated and you probably have a good idea with the training classes. Have a look at www.apdt.co.uk for a list of reward-based trainers. Be careful which trainer you go to - don't go to a trainer who tells you to, say, push your dog's bum down into a sit (should lure a sit with food) and don't go to a trainer who advocates any kind of "corrections", such as lead jerks etc. Might be an idea to go and watch a class before you join.