Very confusing. If such a "theological noncognitivist" was told that Poseidon is considered a god and asked if he believed Poseidon actually existed what would he answer?
Polytheistic gods tend to be like literary characters, they are identifiable and finite. Poseidon is a chap with a trident, son of Cronus and Rhea, brother of Zeus. He has very human characteristics, but also has special powers, a bit like Superman. The proposition 'Poseidon exists', in my opinion, is false. [Poseidon exists, as in a real person with a body that occupies space and came into existence at a certain point in history]
The monotheistic God has no form, is infinite, eternal, onmipotent, omnipresent. God neither exists in space, nor in time, and must therefore have created space and time. He is specifically not human and is said to be beyond human comprehension. Different and contradictory things are said about Him, and his characteristics and actions don't seem consistent with each other.
We can attache certain words to 'God' such as eternal and infinite, but these ultimately make absolutely no conceptual sense to us.
What do we mean by 'God exists'? We already know God is outside of space and time and has no physical form, we also don't really have any idea about what God is, except a few supposed characteristics He has. By what standards can we judge His existence?
The words God and exists cannot be collocated in a meaningful way, like 'emotion is pink' or 'sound is compassionate'. If 'God exists' is not a proposition and is neither true nor false, it is meaningless. This means that commonly used definitions for theism and atheism are also meaningless.
That is what a theological noncognitivist might say.