Given a fluffy’s general diet, I would expect their mouth fauna would be as bad as humans, so licky cleans would exacerbate the infection situation.
angry19’s proposed ‘petra fluffies’ had them with saliva with something like 40% ethanol/isopropyl alchohol content to address this issue.
Thk in “Simple Creatures” proposed a solution - when fluffies give ‘huggies’ to each other, it kicks off a pyrexial (fever) response, raising fluffy’s body temperature to kill off pathogens and stimulate the immune system. Hugging also triggers endorphin and oxytocin release, which have a synergistic effect, thus making fluffies feel better at the same time.
Edit: I just realised, under Thk’s headcanon, fluffies literally give warm hugs due to their raised body temperature.