"The executive power shall be vested in a President of the United States of America."means. Claims that the President has complete authority over every official who appears to be in the executive branch look like semantic ones that turn on the meaning of that sentence. And this seems like something a philosopher of language could help us with.
Certainly, some theorists go in this direction: Stavropoulos, for example, and Dworkin in his way. And Chris Green's paper is definitely pressing a semantic argument. But I'm increasingly doubtful that semantics, rather than pragmatics, is where the action is; insofar as arguments about the Vesting Clause are linguistic in any sense, they're focused on that particular use of "executive power" in a particular constitutional / legal context. And this doesn't seem to be what semantic theories are trying to make sense of.
Even a shift to pragmatics may be misleading, though. If we start from the question of which of various possible constitutional orders we have, right now, then it looks like the focus will be on norms of constitutional validity, which may or may not be amenable to (let alone constituted by) authoritative verbal formulation. We care about particular constitutional sentences because of what they do to these norms. And it's not clear why we would expect a general theory of pragmatic content to help us much with this in any concrete way, as the relationship between sentences and norms may be quite system-specific; and indeed is perhaps the defining feature of that system.
[Many theorists have analogized the norms of legal validity to the norms of a game. Would we expect a general theory of pragmatic content to help us much in understanding the rules of Calvinball? Not really: we need to understand Calvinball first, in order to see how utterances within it fit into our theory of content.]
This isn't to say that these theories don't apply to constitutional texts; they surely do. But to the extent we're pushed towards pragmatic theories of legal content, it will be a very deflationary sort of application. On something like Brandom's inferentialism, for example, it seems like we'd characterize the meaning of the Vesting Clause within U.S. constitutional practice as the commitment of everyone involved to take it as a premise in their constitutional reasoning; in particular, their inferences towards entitlements to "x is un/constitutional"-type propositions. But this formulation clearly doesn't help us make the right inferences of this sort; it just makes explicit what's going on already.
Originalism is a sort of foundationalism concerning constitutional inference: it claims there's only one sort of valid argument towards proposition-entitlement, though others may be good if they are taken to reliably generate the originalist sort. What kinds of arguments could justify this claim? Only ones internal to the inferential practice itself, I think: arguments that demonstrate why non-originalist inferences are somehow defective (perhaps by leading inevitably to contradiction). I think this is the most respectable gloss to put on many of the arguments put forward for originalism; and I think that the pervasiveness of "constitutional faith" is why moral arguments are actually respectable in this context, though only when formulated in a certain indirect fashion. This is also how many dynamist arguments work: one commenter pointed to Stevens' concurrence in Georgia v. Randolph, in which we can see his attempt (ably rebutted by Scalia) to demonstrate why originalist inferences would be unsound (namely, because they would lead to contradictions with commitments to gender equality).
Now, I take the project of constitutionalism to aim at the institutionalization of public rules of constitutional inference. I think this, in fact, actually points towards internal arguments against originalism, arguments based on incentive-incompatibility (at least within a supermajoritarian amendment framework). But, alas, few writers seem to be advancing claims like this.