The delving into metaphysics needs some clarification concerning objections to logic and reason. The premise is the logic and reason are limited to understanding the true nature of existence because it's not possible to give the a priori statement in question because it requires that one uses logic and reason to conclude it. I'm not going to argue against that objection because it is valid, though perhaps not conducive. But I will argue against the Bible being the, or a, gap filler.
The first thing to look at is the premise that logic and reason are limited. What this suggests is that while logic and reason are valid, they are limited to understanding the true nature of existence. But what it does NOT say, and what I would suggest no one would claim, is that logic and reason is invalid. Indeed, in order to have any sort of meaningful conversation, one requires the use of logic and reason to make communication coherent. To accept logic and reason as tools of understanding is to accept that logic and reason are not contradictory, since by definition it necessitates non-contradiction. Therefore, if there is a single exception arising from a separate a priori statement which removes the premise that logic and reason being valid, then logic and reason as a whole is invalid since it exists as a single a priori statement. Since we accept logic and reason to be infallible (though human application of logic and reason is fallible) we must assume that should another premise invalidate the premise that logic and reason are valid, then we would throw out the other premise. For a simple example, think about the premise that addition is valid so that one and one make two. If we suggest a separate premise that gives rise to us concluding that one and one makes three, then we assume this separate premise is invalid, not the premise of addition being valid.
To summarize, I would suspect that we all believe logic and reason to exist and to be valid. At the very least, logic and reason is therefore a subset of our understanding of existence. To find an exception to logic and reason is to invalidate logic and reason. We assume logic and reason to be valid. Therefore, any premise that would invalidate logic and reason is invalid itself, and logic and reason remain.
Now the argument of the Bible comes into play. The premise suggested is that the Bible gives truth to our existence (from here on out referred to as BB) beyond the limitation of logic and reason. If we can find that this new premise leads to a contradiction with the premise of logic and reason, we will assume that the premise of the Bible is invalid.
As it goes, the Bible is in fact contradictory in its literal form and thus would necessitate an exception to the a priori of logic and reason (from here on out referred to as L&R). Therefore we automatically can reject the idea that the Bible gives truth to our existence in its literal form. Thus, in order for BB to hold weight, we must assume that the Bible must be interpreted. As it is, the Bible has been interpreted many different ways that are mutually exclusive, as evidenced by the number of denominations of Christianity. Since they are as whole mutually exclusive, they cannot all be correct and furthermore only one or few (if some small set of the denominations are not mutually exclusive) can be correct because otherwise they would constitute an exception to L&R. Since this is the case, there is a very small minority of people (if any) who interpret the Bible who are correct in its interpretation. But here's the kicker. Roughly at least one person who follows any single interpretation will be just as convinced (or at least it cannot be proven otherwise) as someone else with another interpretation. So how do we know which interpretation is the correct one? In what capacity can we measure which interpretation is correct? There is no answer to this and thus one can never assume to have the one true interpretation. For any a priori statement to be made, it must be clear (i.e. not vague) and the very meaning of the a priori statement would have to be agreed upon or obvious, though not necessarily accepted. Therefore, BB needs some extra clarification, specifically that it would be necessary to transform it to say that "a specific interpretation of the Bible, as written , gives truth to our existence" (from here on referred to as IB).
There's not much to say (that I know of) to someone who considers their interpretation to give rise to the IB in question. All one can say is that there are perhaps an infinite other possibilities which could take the place of any particular IB (including other IBs) and it would suggest a delusion of grandeur for one to admit that only his/her interpretation is the true interpretation, though not invalid. The only way to avoid this is to not accept L&R, which I again would suggest no one ought to or would do. As it stands, the correct interpretation cannot be discovered by other a priori statements that don't have the same problem themselves, i.e. there is no objective means to discover the correct interpretation. Since, then, very few people will agree to accept anyone's particular IB, it doesn't stand as a very strong a priori statement. Moreover, interpretation requires another a priori of subjectivity, since interpretation is specifically subjective. Specifically, one would require an a priori statement which decrees that subjective statements can be decided as true (from here on referred to as ST), for if it cannot be decided as true then no person can claim that their interpretation is correct, which implies that no person may claim any particular IB. ST really opens up a whole floodgate of statements based on subjectivity including but not excluded to: faeces taste good as fact or theism is a good thing as fact (who's to say that these are not true?). From merely an argument of absurdity, it would suggest that ST is not a valid a priori statement, and thus neither can an IB since it requires ST.
To summarize, BB is not sufficient in literal form since it would require an exception to L&R. Thus, BB must be specialized to an IB. IB is not self-evident, and requires ST. ST leads to absurd statements (suggesting an exception to L&R) so we would/should not accept it. Therefore, since IB relies on ST and ST is presumed not valid, IB is not valid. Thus, BB in any form is not valid since it leads to an exception of L&R which no person ought to or would do. Ergo, the Bible does not and cannot fill the gap (if it exists) where logic and reason leave off.