He has a lovely set up there

As you say, as it's a much larger space, he might still be feeling a bit cautious about being out and about. Whereabouts is his multichamber hide? I can't tell!
Sometimes with a nervy hamster, I set the cage up so it's kind of in two halves so they can get used to half the cage (a smaller area) and venture into the other half when they feel more confident. But not suggesting you start changing things just yet while he's adjusting.
How far is his wheel from his house? He might be feeling nervous about leaving the house.
Also I would put his food somewhere else, rather than the wheel or he might just see it as a food dish! I sometimes put the daily piece of veg at the opposite end of the cage to the house (maybe not in your case as it's a very large cage but quite a distance from the house), on a separate dish. They seem to live for their bit of fresh veg and can't resist coming out to get it, so they get used to moving across the cage to get the veg.
I can see what you mean about thinking it may stress him to be out of the cage, but actually the longer they are left without human contact, the more "feral" they become!
I tend to start off with a small area like the bathtub - without doing any handling at all. They soon get used to it. I have a blanket down and a tunnel and hide and a spare wheel, and maybe a small forage box with treats in. They just enjoy exploring a little or might sit in a hide for a wash. By the second time I put a bit of hamster food on the palm of my hand and have my hand flat down/palm up. First couple of times they might avoid the hand but then they get past that and start taking food from your hand, and that is real progress. Also by the second or third time I just start with a stroke on the back with one finger, just once, then maybe once again a bit later.
It can take a couple of months. Maybe being out twice a week. But they get used to it and get to know the tube is their taxi and get used to being transferred to the bathtub.
Once they get used to a stroke on the back with one finger, and eating out of your hand you could keep doing the occasional stroke. And they can start sitting on your hand while eating or putting a paw on it. Once they sit on your hand, you can gently lift your hand up a little bit, so they get used to that, but put it down again straight away.
My last two were at the stage of being able to be lifted on a hand and gently handled for a few seconds before I moved them to a playpen (larger area) and both are now very laid back about being picked up while in the playpen, and both are "trained" to walk into a tube if it's offered
I also once adopted a Syrian who was scared of hands and was a biter. It took two or three months with him and then he was happy to be picked up and held as long as I was on the same level with him.
So I don't think it is just for you

It's also for them, so they gain confidence and learn to trust a human. Which can be quite important in case they ever need to be taken to the vets. It would be very stressful for a hamster who had never left their cage, to suddenly have to be put in a pet carrier and go on a journey to the vets and be handled, if they weren't used to it.
The key is to just start slowly. Even when they have a lovely big cage, they enjoy a change of scene! And curiosity kicks in. They also behave completely differently out of the cage to in it. In the cage they are territorial.
I think after two weeks and five days, it would be a good time to encourage him into a tube (maybe put it over his house door with a bit of cucumber at the far end). Obviously you need to keep both hands over each end of the tube when you're transferring him to the bathtub. They really do get used to it!
My biter turned into an adorable hamster who had a real bond with me and wanted to be out every night free roaming. He never did like being held for very long - he'd sit for maybe 20 seconds for a stroke - but he trusted me and used to look up at me when he was having fun out of the cage, and would always keep coming back to my feet.