We also use natural cleaners in the house. We have eight parrots, and we have to be very careful about what we use in the house, as many cleaners, bleach included, can kill them.
However, the birds don't go camping with us! We use bleach disinfectant wipes to wipe out the refrigerator after each trip, and then prop the door open.
BTW, we also use the bleach wipes on the hose bib for the fresh water connection, saturating the bib and letting it air dry. You never know what happened on that bib before you got there!
Happy camping,
Elizabeth