This resembles a type of sooty mold aptly called the "honeydew eater," Scorias spongiosa. Images for comparison can be found on the linked page. It's most often associated with honeydew buildup from a native insect called the Beech Blight Aphid. (Despite "blight" in the name, it's not a serious pest.)

Since probably any pesticide product used to either reduce the lanternfly population generating the honeydew or to treat the fungus itself may risk the health of the fish (or any frogs, dragonfly larvae, or other animals in the pond), we'd suggest only using plain water from a garden hose to blast the residue off. Scrubbing it gently with something disposable may help to dislodge it. If that approach doesn't remove enough mold, perhaps it would at least dilute the honeydew residue enough to make it less of a food source for additional mold growth. The mold will go away on its own once the honeydew source is gone (as the lanternflies die out for winter). Since sooty mold is not infecting anything and won't harm the tree, its presence is more of a nuisance than a threat. If the birch is suffering dieback, it would be unrelated, and Himalayan birch is not highly heat-tolerant, so mid-Atlantic summer conditions may be taxing its ability to thrive or rebound from stresses like heat, drought, or wood-boring insects.

Miri