Sorry if you already said, but is it the mini duna fun or the large duna fun? The large duna fun (70cm by 40cm) is not a bad size and does have levels, and could be adapted a bit. The mini duna fun is far too small for a Syrian.
He is adorable! But I noticed in your photo that his flying saucer is way too small for a Syrian and that could also be a cause of bar chewing. They sometimes bar chew if something is not right with their wheel or water bottle. It would be extremely uncomfortable and difficult for him running on a flying saucer that is too small - more so than a wheel that is too small because the way they run on a saucer is different, so it does need to be 12". Whereas a wheel could be big enough at 10 or 11" for an average sized Syrian. It looks like a 7" flying saucer which is only big enough for a dwarf hammy (apologies if I'm wrong and it's bigger!) The 12" flying saucer is huge and would take up most of the floorspace probably, so getting something like a 28cm trixie wheel might be a good move and then see if the bar chewing stops.
As it sounds like taming is going well, you could put off upgrading him to a larger cage until he is more tame and settled, if it's the large duna fun, but if it's the mini duna fun I would upgrade now.
If it is the mini duna fun and you need to explain to your parents - then this info might help:
1) There unfortunately is no legal legislation for pet cage size in the Uk - it is a commercial enterprise for pet shops, who can buy small cages for less than £10 and sell them at a massive mark up price.
2) In Germany there is a legal welfare minimum for a hamster of 100cm by 50cm floorspace cage size.
3) In the Uk the RSPCA used to have a recommended minimum cage size of 80cm by 50cm floor space, which is why there are so many 80cm by 50cm cages available. Floorspace doesn't include the floor area of levels, but actual floorspace.
4) The RSPCA no longer have a recommended minimum but say that a cage needs a lot of space and needs to be large enough for roaming space, after the addition of houses, toys and wheels. And needs to be large enough to allow for enrichment.
5) Enrichment is classed as - enough substrate, toys, hidey places, things to do, a level, a house that is big enough to buikd a nest in and is dark inside (ie somewhere to retreat to) which the RSPCA also say is a requirement.
6) It would not be possible to achieve a suitable sized house, wheel, enrichment toys and house, with enough floor area left, in anything smaller than a 70cm by 40cm really (eg something the size of the duna multy or the duna fun large).
7) All pet cages come with a 6" or smaller wheel, regardless of the size of the hamster! This wheel is only big enough for dwarf hamsters, not syrians, and a Syrian runs very curved in a 6" wheel which can cause them back pain (they usually ignore the wheel or start bar chewing).
In the Uk the responsibility to comply with pet welfare legislation is with the owner, not with the pet shops who sell the pets or the cages. The recent Uk pet welfare legislation is a bit vague and woolly, but here it is.
https://www.rspca.org.uk/whatwedo/en...imalwelfareact
The vague and woolly bits are - the need for a suitable environment, and the need to be able to exhibit normal behaviours. This is vaguely covered by their requirements for Syrian hamsters:
This is taken from the link below, with various links to RSPCA recommendations for caring for hamsters
Most pet welfare charities still use the minimum recommended size of 80cm by 50cm in the Uk. Size is not everything though and there is no point having an 80cm by 50cm cage with nothing in it lol. And likewise some owners let their hamsters free roam a lot and their hamster is happy in a not so large cage as a result. But generally, they spend most of their lives in a cage, and are at their most active at night, when in the wild they would run up to 10 miles a night, so a wheel that is comfortable to run in is one of the most important things for their overnight exercise.
If you got the flying saucer at Pets at Home then do get your money back! Take it back and ask for a refund and you can then put the money towards a suitable sized wheel. Although if it's the mini duna fun then a 28cm wheel won't fit, so you would need a cage upgrade, or a 20cm wheel for now. But nothing smaller than a 20cm wheel for a syrian, or a 12" flying saucer.