Mormon concept of 1 Timothy 6:16

I fully agree with @depperm's answer. But just to have fun with the question, let's assume Joseph made no corrections. What could the verse possibly mean from the LDS perspective?

We believe nobody was resurrected before Jesus Christ. From Acts 26:22-23 (see also Colossians 1:18)

Having therefore obtained help of God, I continue unto this day, witnessing both to small and great, saying none other things than those which the prophets and Moses did say should come: that Christ should suffer, and that he should be the first that should rise from the dead, and should shew light unto the people, and to the Gentiles.

Remember, we of the LDS faith believe that our Father's work and glory is to "bring to pass the immortality and eternal life of man." (Moses 1:39) The difference between immortality and eternal life being resurrection and exaltation (see my answer to this question). Since all people will be resurrected but none but Christ had been at that point, it was perfectly sensible to describe Christ as the "only immortal."

(Let me note that many people believe Matthew 27:51-53 describes Christians rising resurrected from their graves. They're interpreting after his resurrection to mean "after the manner of" rather than "sometime later." Since Jesus was the first to be resurrected, and this statement was made in Matthew immediately after Jesus died on the cross---three days before his resurrection---it's obvious to me that it's the later interpretation. But that's an answer for another question.)

As for why we (meaning Joseph) think the verse was mistranslated or lost something from Paul's original writing (which doesn't exist) to today's translations, we need only look at the expanded reference you provide. From verse 14:

...until the appearing of our Lord Jesus Christ...

And verse 16:

...whom no man hath seen, nor can see...

Paul appears to be contradicting himself, suggesting that we are waiting until the next appearance of Jesus... whom no man has seen, nor can see. One of those two statements is wrong.

Worse, he teaches that Christ is dwelling in the light which no man can approach unto. Except that Jesus spent His entire ministry teaching the He is the light and that He is completely approachable. ... Unless maybe Paul was being very literal: perhaps a person in their mortal body can't stand before Divinity without help? Jesus Himself was transfigured to see Old Testament prophets (Matthew 17:1-3) and they weren't even resurrected, yet. From Moses 1:11-14 we learn:

But now mine own eyes have beheld God; but not my natural, but my spiritual eyes, for my natural eyes could not have beheld; for I should have withered and died in his presence; but his glory was upon me; and I beheld his face, for I was transfigured before him.

Without that transfiguration, no one on this earth can stand before the risen and glorified Lord.

So, why didn't Paul just say this? We believe Paul was a better teacher than that. Why tell Timothy that he was called to eternal life (1 Timothy 6:12) and then tell him it won't matter because only Jesus is immortal and no one can approach or even see Him? That begs the question, what's the point of building up this entire house of cards to explain a confusing verse when Paul's usual lesson is about having a hope in Christ and availing oneself of His atonement?

Conclusion: without Joseph's correction to 1 Timothy 6:16 a complicated house of cards is needed to even try to understand and rationalize the verse, which is a better than average sign that the verse was improperly translated. On the other hand, Joseph's repair sounds very much like Paul and conforms to Paul's other teachings.

Finally, our 9th Article of Faith teaches:

We believe all that God has revealed, all that He does now reveal, and we believe that He will yet reveal many great and important things pertaining to the Kingdom of God.

Including repairing misunderstandings in the Bible (and anywhere else, for that matter).