All I can add is, a down is the most vulnerable position you can put a dog into, so although she's happy to do it at home, the reason she isn't happy at training classes is the fact that there are things there that phase her, maybe the bigger dogs, maybe one of the owners, maybe even the trainer (I had that problem when I was puppy training). No dog likes to go down unless they are totally comfortable with their owner and their surroundings, it goes against the grain, i.e,. the fight or flight syndrome, they like to be on their feet! Good luck.

Of course, it's much different with an older dogs who knows you inside and out and has explicit trust in you, but that's not gonna happen with a 6 months old puppy imo, so it's going to take time.

Glad you are seeing improvment
Its really important to rem that dogs dont generalise all that well so once you go to somewhere different you have to expect less, in a class with lots of dogs I would start again as if you were teaching it from scratch, dont mind if you are treating more and asking for less than the other people in the class, make it fun and easy and in no time you will have a happy solid stay

I was looking for vids of mine staying - the only one I can find is agility start line
http://www.youtube.com/watch?v=w1K4CFme-RU
Ben is a worrier who used to like to be glued to my leg, but once he knew what was wanted with a stay he has a happy stay
- he dosent usualy yap as much as that, it was our club fun day, we had all been on the wine and having a total daft time, the dogs were all hyper (the one running in was the judges dog) - hence why my handling sucks and Ben runs off course - something he has never done but we were having fun