Only butterfly caterpillar that eats Rose of Sharon that turned up in my search was Gray Hairstreak. I see them often in my garden, but I never noticed them eating my RoS but they use other plants for larval host too.
Those are tiny butterflies and presumably the caterpillars are tiny too. Do they do that much damage? Rose of Sharon is a huge pollen source and the blooms are constantly visited by bees and wasps that practically bath in the pollen (as well as all different butterflies and hummingbirds too).
Personally, I wouldn't spray with any insecticide -- there's too much risk of harming the beneficial and desirable visitors. I'd let Rose of Sharon sustain considerable damage -- they are tough and will recover.
Besides, it's almost end of the season for the Rose of Sharon. They are starting to yellow and shed leaves here. First to lose leaves in fall and last to break bud in the spring.
