Quickie: Help be identify the correct branch of linguistics

[vegfarandi]

If I say that the presence of a marked oblique argument on a verb is generally lexically determined, how do I describe the mechanism when an oblique argument is marked without the verb lexically requesting/requiring it? For example, the language allows possessors of objects to be marked as oblique argument: e.g. she cut his hair --> she cut him the hair. Is this a "pragmatic" process? "Stylistic"? "Syntactic"? "Semantic"? "Semantosyntactic"? "Syntactostylistic"?

Is there even a word for such a thing?

[Kuchigakatai]

"Syntactic obliques" would be fine IMO, especially because you contrast them with "lexical obliques" (or lexical verbal obliques if you will).

[cedh]

For example, the language allows possessors of objects to be marked as oblique argument: e.g. she cut his hair --> she cut him the hair.

This specific process is called "external possession" or "possessor raising". Here's a paper about it, which states that the process happens "at the syntax-semantics interface", i.e. it is a syntactic process that's often motivated by semantic considerations, and sensitive to semantic details. "Semanto-syntactic" thus sounds correct IMO, although it's probably not a commonly-used term.

"Syntactic obliques" would be find IMO, especially because you contrast them with "lexical obliques" (or lexical verbal obliques if you will).

This seems to be a good suggestion.