Don't know if this is bee-safe, but ...
The best "off the shelf" water based, mold resistant paint I've found is Zinsser Perma-whte exterior (about $40/gal.) It has a short shelf life (settles out bad and gets "clumpy")They also make an interior version, which I rarely buy. Pretreatment/cleaning with bleach(sodium hypochlorite based, highest concentration you can find, usually under 8%) If 7.5% : mix NO stronger than 1 part bleach to 4 parts water. I normally find 5 parts water to be best but I'm usually dealing with really bad mold.. and spraying a whole house. Lower than 1:5 concentrations work almost as well if you don't mind re-wetting several times and taking longer for it to work. Higher concentrations don't to work any better and make things more unpleasant for you.
For really stubborn spots, sealing with "Cover Stain" oil-based after cleaning. It's like "Kilz Original", but rated for exterior use as well.
A "real" paint store may have a mildewcide additive. If you use it, its best to use up the paint because it will shorten the shelf-life a remarkable amount. Pretty sure all of the insecticide additives were taken off the market years back, but read the label closely anyway.(a real shame btw ... excellent product, small amount of insecticide placed exactly where you need it, and no spiders or dirt daubers on your soffets, but stupid people misused it so govt. gets involved.
They might also stock a mildew treatment that works a little differently than clorox and sometimes it will work a little better on some molds.
Killing mold in small areas you don't want to paint ... phenol, that old-time hospital smell, sure to bring back memories if you're over 50 and had childhood trauma in the hospital... but it can get expensive really quick and it's considered hazardous, so be careful.
One that might be of interest here is a borate treatment but it's either a real pain or expensive (if you buy stuff ready mixed). I use it where there's unfinished wood that has a history of dry rot and mold (usually basements, crawlspaces and sometimes attics. ... a mix of ethylene glycol antifreeze, borax and boric acid. You have to use the "concentrated" antifreeze, formerly called antifreeze, not the 50/50 or "ready to use" ... if it's not really thick and green, it's the wrong stuff.
Then you have to heat it (I do it over a fish cooker 5 gal. at a time) to dissolve the solids, cool down slowly, and you end up with a supersaturated solution that you can cut with water to spray or brush ... You can store it in the original jugs for a week or two if you put it in a temperature-stable place
For beehive treatment, I would think surface cleaning (or painting) after soaking it into the wood, would be best. It will discourage/"kill a few" termites but will not affect carpenter bees fwiw.