Copular verbs are very interesting from a syntactic perspective, because they behave just like auxiliary verbs in languages that have auxiliary verbs. For example, they participate in subject-auxiliary inversion patterns in English questions, and they can precede "not", unlike regular main verbs. This is not just true of English, but of all languages that I am aware of. Although the predicate phrase following the copula is clearly not a main verb, it, too, can behave syntactically like a main verb in some languages, e.g. Breton. So I would definitely treat it as a function word in English.
ETA: This topic probably belongs in the morphosyntax forum rather than phonetics and phonology.