Before I try to answer this question, I think I need to give some context as to how Jehovah's Witnesses have arrived at their current beliefs.
The "founders" of Jehovah's Witnesses used this method to interpret the Bible:
“Someone would raise a question. They would discuss it. They would look up all related scriptures on the point and then, when they were satisfied on the harmony of these texts, they would finally state their conclusion and make a record of it.”
What that means is, we don't draw conclusions from the scriptures unless the Bible backs it up, and we don't assume that the Bible contains a specific belief until it can be proven. We don't use one belief to justify another belief; we use the scriptures to justify each belief individually. Over the years, Jehovah's Witnesses have changed, refined, and clarified various beliefs. This isn't because the Bible has changed; it's because different scriptures have been highlighted which bring out new meaning to the previous scriptures which were used to justify that certain belief. Those same scriptures may have also been the basis for another belief, and so that other belief can be refined as well. This process is described at Proverbs 4:18:
But the path of the righteous is like the bright morning light
That grows brighter and brighter until full daylight.
So now to answer your question:
Jesus being Michael the Archangel is certainly a relevant belief, because it shows the love, grief, and sacrifice which Jehovah had to experience in order to watch Jesus, his first, most powerful, and most beloved creation, die and sacrifice himself for God's purpose. This is what makes Matthew 3:17 and Matthew 26:39 such emotional scriptures to read.
If Jesus is actually not Michael the Archangel, it would mean that the scriptures which support that belief would need to be reinterpreted in light of other scriptures. The new interpretation of all these scriptures would then have the opportunity to be applied to all of our other beliefs, and so more adjustments would follow over the years. I don't feel qualified to predict what these adjustments would be.
The identity of Michael the Archangel doesn't seem to be one of the "fundamental" beliefs of Jehovah's Witnesses (which are listed here). The article on jw.org, "Who is Jesus Christ?" makes no mention of the name Michael, so I suppose it's not extremely vital to learning who Jesus is, but it certainly adds a lot of context to why Jesus was an appropriate choice for the Messiah, as Michael was also evidently the angel who led the Israelites through the wilderness (Exodus 23:20, 21), and will also be the one who will stand up "in behalf of [Daniel's] people" during the great "time of distress." (Daniel 12:1)