If he is chewing the bars having just gone in the cage, he's probably panicking a bit. I would pop him back in his old cage, and then let him have another play in the Alaska tomorrow or the day after. With plenty of substrate and a tube or something and maybe a familiar toy from his old cage. What cage is he in at the moment? Is it barred or a tank style? (Assuming it's barred).
As for a wheel - 6" is definitely too small and yes may well be the reason he's not using it. As Souffle says you could get an 8" or 8.5" but a popular one is the 11" Trixie wheel and it would fit well in the Alaska and isn't too expensive. If he's quite chunky a bigger wheel might be better and I think they run better in the bigger size.
It's currently on at £8.99 on Amazon (often between £9 and £11 - the prices fluctuate).
When he's had a play in the cage a couple of times it'll be more familiar. Then when you move her over you can keep all her old substrate from the old cage and spread it out on top of the new substrate in the Alaska (you'll need more substrate in the Alaska first, with it being bigger) and move all her other toys/house etc across and try and keep them in similar locations. Also don't be tempted to clean things before moving them over
They need to still smell familiar. He'll need a couple of weeks to settle in again but you could offer treats through the bars and take it slowly.
The 12" flying saucer is also good, as an extra, but I wouldn't use it instead of a standard wheel as they run better (straighter) in a standard wheel, but if you've got room for both I'm sure he'd love it!