White Vinegar. It disinfects just as well as bleach and is nontoxic as well as cheap. Before someone pipes in with some comment about the smell I'll preemptively ask this, do you like the smell of bleach? If your great grand mother didn't use it then it probably isn't safe.
I would argue that bleach is much better at killing microbes than vinegar....My great grandmother died of an infection that could have been prevented with modern disinfectants.
If you're only cleaning, why waste the expensive stuff like chlorhex? Dish soap works great, and soap and water gets rid of 99% of bacteria. If you want to disinfect, follow up with a disinfectant, but be sure to pay attention to contact time. Chlorhex won't do anything for disinfection if you don't let it contact the surface to be disinfected for 10 minutes.
Would you breed a wobbler dog even though it had a cool coat?
I soap and water the take all up, then rinse it out and use Chlorhexidine...
I also leave the snakes "ornaments/furniture" as it was before so the snake couldn't possibly go into shock from too much cleanliness?