I have two litter boxes and the sides are about 5", give or take a little. The smaller of the litter boxes has NO lip on it and they hop in/out quite easily......both being adult cavies. I began by using disposable litter boxes with a doorway cut in it. As they became accustomed to the litter box, I gradually raised the height of the "lip".
As an aside, I too absolutely loathe shavings of any sort. If you have male cavies, they have a nasty habit of getting caught in the penile sheath or in the anal sac and causing problems. I'm very fond of
Planet Petco Small Animal Crumbled Litter
https://www.petco.com/product/114718/Planet-Petco-Crumbled-Paper-Small-Animal-Bedding.aspx?CoreCat=OnSiteSearch It is superb at absorbing waste, controls odor, and is quite reasonably priced. The only drawback is that it has baking soda as a component and the dust from that gets all over my drapes. Small price to pay.