Yes it is a big cage to fill. It sounds like you need to be getting bulk bedding so you can use loads and it's cheaper. Carefresh isn't cheap and doesn't go that far. Are you in the Uk? I use Fitch - which is popular on here. A 10kg bag costs about £22. I use about a third of it to fill a 100cm cage about 5 to 6" deep. After that it lasts ages with spot cleaning mainly.
Your cleaning regime sounds fine but it does sound like she could do with more substrate generally.
She sounds very active - a destroyer! Is her cage in your room? ie is it keeping you awake at night as that can be stressful.
It does sound unfortunately, with her being a chewer, that she will chew a wood cage and just see it as something to chew. She might give up eventually if she can't escape and it's better than chewing bars. The only alternative would be a glass tank or a diy cage with melamine board. You can probably just keep patching this one up if she chews a hole. But if it gets too bad maybe you could "line" it with melamine board on the sides and back. That would mean removing the shelf but you could always stick some legs on it and have it as a standing shelf.
My instinct would be to get her a good big wood house. Although she'd probably chew that as well!
I had this one for a while for our Syrian - your cage is big enough for it. It's described as a rabbit house but nowhere near big enough for a rabbit. It does have quite a large door though and is quite tall so it works quite well if you stand it directly on the base of the cage, rather than on top of the substrate, and put some substrate inside it (as if it's standing on substrate) and then just pile the substrate around it on the outside, so it;s part underground basically. This also reduces the sizeof the large door. If you have a bendy bridge tunnel, you could put that over the doorway - makes it dark inside, and gives a tunnel entrance, and also makes a ramp up to the roof. The roof lifts off so you can check inside without having to take the house out and disturbing her nest.
I expect she will move into it, if it's dark inside, and build a really big nest in there. That should keep her busy.
Better to use plain white toilet paper torn into strips, than kitchen paper. Kitchen paper isn't as dissolvable if swallowed and toilet paper is cheap enough. They like it too. She may pouch some to take to her nest.
The only thing with a wood house is it can pee stained. A lot of people paint the inside of wood houses with plastikote - I'll link that. I managed without doing that for a long time because our hamster uses a litter tray which fitted inside the house. Also the house fits together like a jigsaw - the pieces slot together. So I'd just scrub a piece under the hot tap if it did get pee on and that sorted it!
If you give them a large house/nesting box they almost always move their toilet corner inside the house, so putting a litter tray inside it usually works. If you put it in the back corner opposite the door it's the best place as they tend to build the nest furthest away from the door at the darkest end. Corner litter trays like the one linked below, fit in fine. Chinchilla bathing sand in it is best - a tube of that lasts ages. Then you just take the roof off the house and empty the litter tray every few days and the rest of the cage stays clean and dry a lot longer
I don't worry about poops really unless they start taking over.
Ferplast Sin house
https://www.amazon.co.uk/Ferplast-46...r=8-2-fkmrnull
Corner litter tray
https://www.amazon.co.uk/Hamster-cor...gateway&sr=8-6
Chinchilla Bathing Sand
https://www.amazon.co.uk/Supreme-Pet...s%2C161&sr=8-1
I'll link the Fitch bedding as well. The 10kg one suits me. They also do a 20kg one which works out cheaper again but £35 seems a lot to fork out all at once and it's quite big so you need somewhere to keep it. It;s just recycled food grade paper - looks a bit like Carefresh but I think it's nicer and no dust.
https://www.fitchfirst.co.uk/fitch-p...-delivery.html