I think I would stick to a wire cage if that's what he's used to. The Alaska cage also has a nice big front door which helps with interaction, and a shelf for hiding under.
I think taming in the cage could actually make him more skittish unfortunately. It's something about having a hand in their cage! It's their territory and it can freak them a bit even if they accept "the hand". Charlie follows my hand around anxiously if I have to do something in his cage, checking I'm not going to steal anything lol.
You can set up a cage so there are things to sit under. Our hamster seems to need this too - he used to be incredibly nervy for the first year or so - his favourite places to sit in his cage were under his sputnik or under a little shelf.
The good thing about a cage (especially one with a big front door) is you can use the roof for hanging toys and the bars to attach ledges and shelves, which gives them overhead cover and also an extra kind of run higher up. I tied a rat tube to the roof of my cage, leading between the sputnik and the shelf so it actually gave head cover at that level too and gives a kind of circular route - he can go up one end, across, down the other side, and back to his house at floor level and the things above them make them feel a bit safer I think.
So although a bin might make it feel more enclosed from the sides, it isn't as easy to stop them feeling too exposed from the top.
Taming can take time - and they usually need something to hide in when they're in the bathtub - a large rat tube is good as they can sit up inside that - a cardboard box or something also. I think their main motivation when out of the cage is to find a dark spot to sit! But sometimes it depends on time of night as well I think. If they're a bit sleepy or dopey they just want to hide and could even fall asleep. I used to hide little treats in the toys in the bathtub and Charlie would forget about being nervous then.
Once he is in a bigger cage, his behaviour could change as well. It can take a couple of weeks for them to adjust to the new environment and when I've done it with Charlie he explores excitedly for the first day or two, then goes a bit reclusive, but after about 10 days he's quite chilled and in a routine and using all of the cage and moving things where he wants them lol.
I haven't used a bin cage personally. But Charlie's first move was the other way round - from an enclosed perspex thing (3 tier rotostak) to a large cage. He did get a bit nervy at first and seemed to be scared of feeling the air through the bars. But he soon settled in and then he was having a whale of a time jumping off his wheel into a big pile of substrate
So maybe I'm biased but I think the easier interaction with a cage would be better. You can have a cage so there are some sheltered parts - one member on here had some hemp mat on the sides in one corner of a shelf which made a kind of enclosed den area.
I don't know how small the current cage is, but if you moved him you could maybe put the old cage inside the new one and let him explore gradually from there the first few days - or at least the cage base - you can set the cage up around it and fill up the centre area with an extra tunnel or something later.
He does sound like he might be a shy little boy but he may gain in confidence in a different environment.
I found upgrading set taming back a bit as well - we had to wait a couple of weeks after the cage change to let Charlie settle and then kind of start again, but it was a bit easier second time round because of the lovely big front door on my old cage, like the one on the Alaska. My cage was on the floor and I could just open the door, put a floor cushion on it and Charlie could just walk in and out easily.
You'll get there. It depends what you think of as tame as well. I think of it as when you can hold them and they don't panic lol. Even after Charlie was tame he is still quite independent and won't sit on you for long - a few minutes maybe then he wants to go off and find somewhere dark to do his own thing.