I'll offer a response in 2 parts:
- A Latter-day Saint's perspective
- A historian's perspective
A Latter-day Saint's perspective
It was a similar concern that led Joseph Smith to start asking questions that the ministers of his day could not answer. In his personal history he recorded some of his feelings as he lived through the Second Great Awakening:
8 During this time of great excitement my mind was called up to
serious reflection and great uneasiness...so great were the confusion and strife among the
different denominations, that it was impossible for a person young as
I was, and so unacquainted with men and things, to come to any certain
conclusion who was right and who was wrong.
10 In the midst of this war of words and tumult of opinions, I often
said to myself: What is to be done? Who of all these parties are
right; or, are they all wrong together? If any one of them be right,
which is it, and how shall I know it?
11 While I was laboring under the extreme difficulties caused by the
contests of these parties of religionists, I was one day reading the
Epistle of James, first chapter and fifth verse, which reads: If any
of you lack wisdom, let him ask of God, that giveth to all men
liberally, and upbraideth not; and it shall be given him.
12 Never did any passage of scripture come with more power to the
heart of man than this did at this time to mine. It seemed to enter
with great force into every feeling of my heart. I reflected on it
again and again, knowing that if any person needed wisdom from God, I
did; for how to act I did not know, and unless I could get more wisdom
than I then had, I would never know; for the teachers of religion of
the different sects understood the same passages of scripture so
differently as to destroy all confidence in settling the question by
an appeal to the Bible.
13 At length I came to the conclusion that I must either remain in
darkness and confusion, or else I must do as James directs, that is,
ask of God. I at length came to the determination to “ask of God,”
concluding that if he gave wisdom to them that lacked wisdom, and
would give liberally, and not upbraid, I might venture. (Joseph Smith
Joseph Smith recognized that the same passages of scripture could be argued to support a wide diversity of doctrines. I believe this is, fundamentally, the same observation that stands behind the OP.
His response, inspired by the Biblical text, was to seek further guidance from God. The subsequent verses in his personal history describe his First Vision, and the resulting Restoration of many Gospel truths.
The experience of Joseph Smith demonstrates several additional sources that can help clarify Biblical passages that might otherwise be interpreted several different ways:
- Additional scripture that has come through the Restoration. In the context of the debate in the OP, Alma 40 and Doctrine & Covenants 138 unambiguously affirm post-mortal consciousness.
- The teachings of living prophets. We believe the Lord's church is led today by apostles & prophets who are every-bit as authoritative as Moses or Peter. For a brief, relevant example regarding post-mortality see this sermon by Thomas S. Monson
- Personal revelation. The power of the Holy Ghost is available to all who will seek it. The Gift of the Holy Ghost comes by entering into covenants executed with God's authority. Two of the features of this gift described in scripture are "that he may pour out his Spirit more abundantly upon you" (Mosiah 18:10) & "that they may always have His Spirit to be with them" (Moroni 4:3). I believe in a God who can & does answer questions, and has affirmed to many individuals what He has taught through His modern prophets: people are conscious and active in the spirit world.
A historian's perspective
The degree to which we are dependent on non-Biblical sources in order to understand Biblical texts is difficult to overstate. We have a Strong's Concordance for Greek because of the millions of words preserved in pagan Greek writings. We understand the context of the ministry of the apostles because of the writings of Roman historians.
- If we want to make a text say anything we like, we should ignore historical & cultural context
- If we want to understand what a historical writer actually meant, enlightenment from other sources is indispensable. We cannot understand an ancient text in a vacuum.
Second Temple Jewish history
The records of the culture, customs, and beliefs of the Jews, especially proximate to New Testament times, are a wealth of information regarding how the people who originally wrote & read the New Testament documents would have understood them (we have far fewer such records prior to the 2nd temple era).
In the context of the debate referenced in the OP, 2nd temple Jewish sources demonstrate a belief in post-mortal consciousness. The parable of the rich man & Lazarus in Luke 16 is an example of a teaching regarding post-mortality that is saturated with beliefs prevalent in contemporary Judaism.
Early Patristic Writings
The writings of the Ante-Nicene Fathers represent an era when the church was not yet politically powerful. As a result, people who were already putting their lives on the line to practice their beliefs didn't have incentive to keep an Emperor (or any other political figure) happy by compromising on what they believed. These writers are within just a few generations of the apostles. They disagreed with each other all the time. But where there is a clear majority (and occasionally they're bordering on unanimous) view, this ought to be taken seriously. Did all of the disciples of all of the apostles completely lose the Gospel message that quickly? If so, why do we trust that they handed down the New Testament to us accurately?
In the context of the debate in the OP, the Ante-Nicene Fathers support post-mortal consciousness by a significant super-majority.
Although I take no issue generally with the idea that scripture interprets scripture, I suggest the maxim can be overdone if taken to an exclusionary extreme. All manner of sources enable us to understand scripture. We may not treat them all of these sources with the same level of authority or reliability, but we ignore them to our own loss.
A source I particularly treasure is modern revelation. I believe in a God who speaks in our time, to our knowledge base, and for our understanding. His revelation in the past is of inestimable value, but it was not fundamental to our salvation that we learn Hebrew, Greek, or history. I believe His ongoing revelation today is the most directly applicable resource for resolving ambiguity in ancient holy writ.