First of all, I would say to answer your question the same thing as succintly formulated in a comment here by you:
[...] exist is not transitive. That seems like a succinct way to answer what I was asking. So the syntax of a verb of existence is different than copulative which is transitive. They are mutually exclusive.
That's my understanding, too.
However, if we allow us to go away from the argument-taking syntax of the verb (where the argument is the direct/accusative object), i.e., look for answers that convey the same meaning as you are asking for and use the verb "exist", but with a different syntax, we might find some constructions which are not-so-bad grammatically in some languages:
? I exist here as an assistant.
or Russian equivalent:
? Я существую здесь как помощник.
I exist.Prs.1Sg here as assistant.Sg.Nom
? Я существую здесь в качестве помощника.
I exist.Prs.1Sg here in quality assistant.Sg.Gen
Here the additional phrase plays a role close to an argument to "exist": it's a "tight" member of the verb phrase, but syntactically expressed differently from a direct/accusative object -- as opposed to, say, a more "loosely" apposited comparison phrase (
I fly like a bird or
I fly like a bird flies in English; in Russian punctuation, there is a rule about the conjunction
как "as" which says that if the phrase introduced by
как is argument-like, one does not use a comma, as in my examples above, but one should use a comma in the second class of examples:
Я летаю, как птица "I fly as a bird).
So the feeling is that such sentences are close to what you are asking.
Or the second predicate can be expressed as an adveb or an adverbial phrase or a small clause (my examples below are best described as having a small clause inside; I think you can try to invent something similar with the usage of adverbs yourself):
?? Я существовал усталым.
I existed tired.Sg.Instr
"I existed being tired"
?? Я существовал усталый.
I existed tired.Sg.Nom
"I existed being tired"
Although this sounds not nice in Russian, but the syntax is more-or-less OK--compare with such sentences with a verb other than "exist":
OK Я пришёл домой усталым.
I came.Pftv to_home tired.Sg.Instr
"I came home and was tired"
OK Я ходил усталый.
I wandered.Impftv tired.Sg.Nom
"I wandered while being tired"
and as you see in the English translation, this syntax explicitly introduces a second predicate in semantics (with the
COPULA-ing in the English syntax); the semantics is the logical conjunction of "exist" and "be tired".