I do NOT recommend using chlorhexidine (or for that matter, the Natural Chemistry spray) for cuts & other injuries to your pet- that's NOT what it's for.
I DO suggest using a product called Vetericyn ointment for topical use in reptiles- it's carried by some stores like Petco, & also online sources of pet products. It's made FOR reptile use, so it's a water-based ointment that won't harm reptile skin. (It's NOT petroleum-based, aka vaseline-based- which tends to mess up future shedding- as happens when using first aid ointments made for human use, such as Neosporin or polysporin; those can be used briefly in a tiny localized amount in an emergency, but they're just not the best option. You can also use a small amount of povidone-iodine, aka the brand name Betadine- topically- for temporary use on a snake wound that's not deep- but again, Vetericyn is recommended & is the best choice.)