I disagree that it’s second-order in the first place. Consider:
“Drinking mercury is unhealthy for children.” and;
“Drinking mercury is bad for children.”
These not only have the same exact surface form, but I would say the same logical form. Both sentences predicate a property of an action vis-a-vis a kind of subject. Both are grounded in facts about what damages or fails to support the proper functioning/flourishing of an organism. If “unhealthy for” is uncontroversially first-order (a real predicate ranging over how an action affects an organism’s biological functioning), then I would say that “bad for” tracks the same kind of fact. But in that case, I can see there’s no principled logical reason to declare that we have moved from first-order to second-order when we move from “unhealthy” to “bad.” Rather, the shift looks motivated by something other than logic; namely, a prior metaphysical commitments (be they Humean, expressivist, or Kantian) i.e., that “moral badness” must be sui generis, action-guiding, and detached from descriptive/functional facts in a way that “health-badness” is not. That’s a substantive metaethical thesis smuggled in under the banner of “what logic requires,” as far as I can tell.
This seems right in line with Foot or Anscombe. If “good” functions attributively (e.g., good qua knife, good qua liver function, etc.) in “non-moral good” cases (without anyone reaching for second-order machinery AFAIK), then it seems to me that the burden of proof for such a variance is on the philosopher who wants moral “good” alone to require an operator-treatment. They need an independent argument for why “moral goodness” is metaphysically discontinuous from this general pattern, and what “moral goodness” is (and so a defense of that notion). An appeal to “that’s how deontic logic regiments it,” is question begging, since deontic logic was built to formalize “oughts,” with those very metaphysical assumptions in mind in the first place. Yet its adequacy as a model for “good” is what’s in question.