You and I can agree that some things are morally valuable.
And I take it by 'inherent' moral value, we mean moral value that does not derive from the thing's usefulness as a means to securing some other end, but moral value that it seems to have in its own right. That is, when we say that something is inherently morally valuable, we mean that it is morally valuable for its own sake, or something like that.
Well, we can agree that some things are morally valuable in that way too. And nothing in the argument I gave implies otherwise. Something will be inherently morally valuable when the subject values it for its own sake rather than for some other sake.
So, the existence of things possessing inherent moral value does not constitute a counterexample to any premise in the argument, so far as I can see.
But perhaps you mean by 'inherent moral value' something more than I said above. Perhaps when you say that something is 'inherently morally valuable' you mean, in addition to being valuable for its own sake, that it is 'objectively' morally valuable.
Okay, but now you have begged the question. Whether inherent moral value is subjective or objective is the issue under discussion, so one cannot just assume it is objective at the get go.
Note, my argument does not assume that moral value is subjective. It's subjectivity is asserted in a conclusion, not a premise.
My argument assumes that for something to be valuable in any sense is for it to be the object of a valuing relation. And my argument assumes that only subjects of experience - minds - are capable of adopting attitudes towards things.
An objectivist must either deny the first premise or the second. Note, to deny the second what is needed is an example of a genuine valuing attitude that is not being borne by a subject of experience. You cannot use moral values as your example, as that's question begging - it is to assume moral objectivism is true, not show it to be. We need an example in which a) it is clear we have real valuing going on, and b) the valuer is not a subject of experience.
Now, I don't believe there are any such examples. But perhaps there are, I am just unaware of them. Plus it does seem that my reason, anyway, represents anything that is not a subject of experience to be positively incapable of having any real attitude towards anything.