My other cage has a dirt ground also and I have never had any problems. I figured they were smart enough to not eat it before I caught them...
There are lots of white spots in that dirt. If those are stones or fertilizer pellets, those will be probably be the first thing the chams try to eat. I don't think they're smart enough not to eat them, and you probably aren't quick enough to catch them before they do.
(I keep a screen cover over my larger pots, and a while back I removed the screen for cleaning - it was off for only an hour or two - and there was gravel in the cham's droppings the next day).
As long as the soil they can get to is clean, organic soil that doesn't contain fertilizers or other chemicals, and is a fine soil composed of small particles (no stones/clods), then it should be fine. Jim Flaherty was commenting here a little while ago that most chams seem to have some sort of biological need to eat soil, so maybe it's not so bad to leave some soil exposed in the cage.
But without ensuring that the soil is safe for your cham, there is a risk of impactions/poisoning caused by ingestion of the soil.
So just be careful. Lay down a generous (thick) topsoil of healthy soil over the floor.
I can't think of many practical solutions to your setup. A screen cover would allow insects and droppings to get through and would make cleaning difficult. And it would be difficult to put a tray of some sort down with those vines planted there, plus there would be a risk of insects getting under the tray and hiding away to prevent them from being eaten.
Maybe a plastic lining of some sort? Either way it would be difficult. Just go for the safe soil...
