I wouldn't go this far. I believe it's probably not a big deal and I've used it for cleaning poop and such, but the safe for animals thing was more of a marketing campaign since they donated it for cleaning birds(which I've also heard didn't do much and many of the birds died either way sadly). A reptile with sensitive lungs won't handle things the same as a larger bird/mammal.
The best thing you can do is not soak it with anything. I'd use vinegar and maybe baking soda if you must for natural branches(I just hose branches off personally, could also let them sit in the sun to uv sterilize). Those two would also do a good job of removing chemical residues leftover on anything artificial you buy.
I would also not use a hammock as their claws get stuck in them and break off. They are not made for chameleons.