Think of it this way. An educated agnostic who is seeking the truth knows that the Bible teaches that Jesus died for their sins. But they also likely know that the Qur'an teaches Jesus wasn't crucified at all. Specifically it teaches that regardless of the reports of such, Jesus was in fact taken up to heaven to be with Allah instead. It also teaches that while the Bible has some authority, and is indeed a good book for learn from, it has been superseded by the more recent revelation contained within the Qur'an. So the agnostic (agnostic theist in particular) who seeks to be "right" with God is put in the rather awkward position of trying to determine which set of revelations to embrace.
That's an interesting distinction between 'agnostic' and 'agnostic theist'.
Epistemological agnostics these days generally are weak ontological atheists. They don't actually believe in any God, even if they don't totally deny the possibility. (I fit there, in some of my moods.)
But yeah, there are agnostic theists as well, individuals who do believe that a God exists, but believe that they personally, or mankind generally, lack cognitive proposiitonal knowledge of that God. At times in history, these represented the majority of agnostics. (Of course, the word 'agnostic' hadn't been coined yet.) We encounter agnostic theists among the deists and among a certain kind of religious mystic.
What I am getting at with this thread is something like this: what sort of assurance can the agnostic theist find in the act of embracing one over the other that allows them to feel certain (or close to it) that they've chosen correctly? If it actually doesn't matter which one they choose as long as they genuinely open their heart to God, then the question is kind of moot. But if it does matter, then the question is an important one.
The deists accepted natural theology but were skeptical about whether any special revelation was true. So they accepted the abstract philosopher's God: first-cause designer and so on, but they questioned the literal truth of books like the Bible and the Qu'ran.
So your problem wouldn't have been felt with much force by the deists, since they were skeptical of all revelations by definition and weren't all that concerned to locate a supposedly one-true-one.
The religious mystics generally arise from out of a particular religious tradition and they typically absorb respect for their tradition's scriptures simply by default. Their agnosticism is usually religious-inspired, inspired not by a lack of religiosity but by an uusual abundance of it, resulting in their elevating God so highly in their thinking that God becomes totally transcendent, above finite and imperfect human conceptualization entirely. That in turn creates kind of a blow-back that puts the literal truth of the scriptures into question and often motivates analogical and allegorical readings.
So I don't think that most of the mystics would have felt the religious choice issue very strongly either. They were already too committed.
There are exceptions though, mystics who have promoted universalist ideas that all religions point towards the same goal and sometimes the idea that there is some perennial philosphy implicit in all religious traditions.