Hi there. There are a number of reasons for bar chewing. It is trying to get attention, but sometimes they're trying to tell you something - eg if something in the cage isn't right for them. Say if the wheel is stuck and not spinning, or something is broken and they can't get to the food bowl, or there isn't enough substrate or something. Quite clever little things when they want to tell you something
It can also become a habit, bar chewing. I'm not sure how your cage is set up now, but the Alaska cage is a good size for a Syrian (although some do like bigger). One thing that makes me think is - the house it comes with (the little white one on the shelf) is not a suitable a house for a Syrian - it's a bit too small, the holes in it are a bit too small (a Syrian could get stuck in them with full pouches) and it is on a shelf rather than open to the substrate underneath.
If the flying saucer is too small (ie smaller than a 12" one) she will be unhappy about that too. A lot of new owners only put a thin layer of substrate on the bottom of the cage which can also make them a bit unhappy. And with toys with chains she might feel insecure as well possibly.
So - without seeing your set up, I would try to adjust it before thinking about trying a different cage.
1) A minimum of 3 to 4 inches depth of substrate is usually recommended. But 6 to 8 inches is best. You won't get 8" in an Alaska without it falling out of the bars, but you should be able to get about 6" in if you fill the base to within 1cm of the top of the base. You can also heap it up in the middle a bit. Hammies love lots of substrate. They can burrow down in it, even dig tunnels. Also it cushions falls if they climb to the top of the cage and then drop. They can really injure themselves if they fall and there isn't enough soft bedding underneath. Deeper bedding also helps reduce the fall height from the top. It can help to have any hard floor toys under a shelf.
In fact if the house is too small and there's not enough substrate to burrow down in to nest, it could be why she's chuntering.
2) House. They need something big enough to build a good sized nest in, that is open underneath, sat on top of the substrate and dark inside. They can then do normal behaviours like burying the hoard under the nest and so on. A shoebox house is a good temporary (or permanent) option. Cut the bottom out of the shoe box, cut a hole in the front for a door and keep the lid as a lift-off roof. That way you can check inside without destroying her nest. A good place to put it might be partly under the shelf. The roof can also make another level. Putting a bendy stick bridge over the door helps make it darker inside and gives a ramp up to the "roof".
3) Wheel. A 28cm Trixie wheel is a good basic, cheap Syrian sized wheel - costs about £8 or £9 from Amazon. A really good wheel is the 12" Silent Runner - you can get bar attachments and fix it to the bars as high as you want it without having to worry about it falling over, being unstable etc, and still have a good depth of substrate underneath it, so that increases the floor area bit too. However, it's an expensive wheel. Costs about £25 and the bar attachment is another £5 approximately. But it's worth investing in and can be used in any cage taller than 35cm - either on its stand (which is metal, safe and sturdy) or attached to bars. A flying saucer as an extra wheel can be fun for a Syrian, but they do really need a standard upright wheel as the main wheel - they run better in it and they need to run with a straight back to avoid spinal problems. Having both can take up quite a lot of floor area, but if you already have the 12" flying saucer I think you could fit both in ok. If the flying saucer is smaller than 12" then it needs taking out (but maybe not until you've got a replacement wheel - they go frantic without any wheel at all).
Yes toys with chains can be an issue - they can get a toe or foot caught in a chain and end up hanging by a broken leg. If it's a tube you could use the chain to tie the tube to the side of the cage though, where the chain itself isn't accessible. Or you can use sisal string to tie things on (it's supposed to be safe to chew but other string isn't) - or I sometimes use cable ties.
So if you get her the basics sorted - wheel, house, substrate and masses of nesting materials (strips of torn up sheets of plain white toilet paper is safest and they like it), maybe scatter feed her food for a while (put half the usual in the food bowl and scatter the other half on the substrate so she can forage for it) she may settle down in the cage. As you need to get a wheel (and maybe a house) anyway it could be worth a try first, because they can be used in another cage.
Other than that, some kind of tube/floor toy is usually popular. Little hidey places like a coconut shell with a hole in (make sure the hole is big enough - 6cm diameter or more), treat chew sticks, are good and also help keep the teeth worn down.
For hanging toys I think rat sputniks are good - they're safe and a good size for a Syrian. Having it hanging next to a shelf helps so they can get in it. I also tied a rat tube to the roof of my cage leading from the sputnik to another shelf on the other side, which gave a kind of roof run.
To answer your other questions:
Pine shelves are ok (untreated plain pine). Most people use plastikote water based enamel paint to waterproof wood items - it's pet safe. But not the spray - the little pot you paint on with a brush.
1 cm square mesh is usually used for bin cages. Lots of ventilation. Elmer's school glue is safe.
If you have a photo of her cage it might help with suggestions
Sometimes it can be something really simple like the position of a water bottle or something.
They will definitely try and get attention if there's no water in the bottle for example.
Anyway hope this helps. It would be hard to find a bin with as big a floor area, and the Alaska has a lovely big front door which helps with taming and interaction.