Is 'the course' a complement or an adjunct in 'Stay the course'?

Neither is correct. "Stay the course" is an idiomatic expression and therefore the verb "to stay", as far as its use in that expression, is not classified as a transitive verb or an intransitive verb (ref.); in consequence it is not possible to say whether "the course" is an object or an adverbial.

Addition

Here is a complement to consolidate the notion that the grammatical analysis of idioms is not really possible. It is contrary to the notion of compositionality (ref.).

Compositionality is the key notion for the analysis of idioms. This principle states that the meaning of a whole is constructed from the meanings of the parts that make up the whole. In other words, one understand the whole if one understands the meanings of each of the parts.

This is what can be read in a pdf on idioms available in The International Journal of English Linguistics.

They are expressions, not subjected to analysis, only some syntactic changes may be carried out in them. Moreover, the diagram from the same source shows clearly the non grammatical approach on the left for the idiom and the usual grammatical analysis on the right for the "regular" meaning of this phrase.