Hello and welcome

I can see you've had lots of great advice above. I agree with Elusive that it would be better to replace the digging box with a large house. I don't think a multiroom house would fit that well in an Alaska - they fit better in 100cm cages. So for now a shoebox house would be ideal. Basically something that's dark inside and has plenty of room to build a really large nest and fit a corner litter tray inside.
If you have a shoebox, you just cut the base out so it's open underneath, and keep the lid as a lift off roof - so you never need to remove the house or disturb the nest. And cut a hole for a door - the best place for the door is on one of the long sides - ideally at one end of the long side, so it's darker at the other end - they tend to build a nest in the darkest place. So the end opposite the door is an ideal place to put a corner litter tray with sand inside. Syrians will virtually always use an ensuite toilet! And then it helps keep the cage clean and dry and you just need to lift the roof off to empty the toilet every few days. It means you don't need to do regular clean outs and can just spot clean as and when needed (which avoids stress for the hamster).
If you have a bendy stick bridge, then putting that over the door works well as it makes it even darker inside and also provides a tunnel entrance (which tempts them in) and a ramp onto the roof. You can put the odd thing on the roof as well - eg a food dish.
At 12 weeks he is not technically a baby but still a bit of a toddler or young child!
I think once he has a good house, his normal habits will kick in - eg nesting, burrowing down in his house, hoarding and burying his hoards in the house. Then he might stop hoarding in the wheel!
I also wonder if the 33cm trixie wheel is a bit large for him to push. The 28cm one is fine for most Syrians, so if you could get one of those, you could keep the 33cm one for a playpen or out of cage wheel until he's fully grown. They can keep growing up to about 9 months old but have done most of the growing by 6 months old.
If he's stuffing bedding near the wheel as well you'll need to keep checking it's clear daily. But again once he has the house he will probably stop doing that as he won't be nesting under the wheel any more. They usually move straight into a house overnight, if it's large and dark inside. But it can help to have a trail of sunflower seeds leading to it and inside, and a smelly treat inside to tempt them to go in and check it out. Eg cucumber.
Actually this house is quite good. I'm using it at the moment for a baby Syrian. It's not a full sized multiroom house but it still works well - and might be easier than a shoebox house. But you would need to get the legs/stilts with it to support it underneath. Maybe 15cm legs if you have 6" of substrate in that area. Or you can get 10cm legs if you want the house partly under the substrate or if there is not quite 6" in that area.
I have it with just the one inner divider at the front in front of the door, a larger compartment behind that with the toilet in and the left hand side just open without a divider (where he sleeps). It's a good house and will fit in the Alaska.
Small Slot Together Wooden Hamster Chamber House Made from robust materials Assembled without glue or nails Flat packed to save on packaging materials Modular, allowing you to change the design to suit your requirements Chambers can be changed into two main configurations Removeable roof for...
www.happyhenryshomes.co.uk
The stilts are in 15cm, 12cm and 10cm. The 12cm ones might be about right for just under 6" of substrate at the house end, but the more substrate you have in - the better.
15cm Stilts for Hamster Houses & Sandboxes Solid Birch Ø25 Dowel 15cm overall length Slot 2cm Deep, 6.2-6.5mm wide (to allow for plastikote or paint) Height of slot from base of stilt 13cm (approx. 6 inches) Cross Slot allows positioning at corners or along sides Order exact quantity to suit...
www.happyhenryshomes.co.uk
12cm Stilts for Hamster Houses & Sandboxes Solid Birch Ø25 Dowel 10cm overall length Slot 2cm Deep, 6.2-6.5mm wide (to allow for plastikote or paint) Height of slot from base of stilt 10cm (approx. 4 inches) Cross Slot allows positioning at corners or along sides Order exact quantity to suit...
www.happyhenryshomes.co.uk