In general, there are no essentials in which Southern Baptists (or those who used to be SBC) disagree. Virgin birth, resurrection...etc.
There are exceptions, of course, to this, but the vast majority wouldn't even question it. The fact that a few rogue places have decided otherwise is not a threat, especially with the Baptist model of autonomy.
As far as inerrancy goes, it's a brittle, broken word that has been used, in many cases pejoratively, to further elevate the Bible. In fact, it wasn't really a common term for that description until the 70s, if I remember correctly.
Unity doesn't mean we agree on everything, especially for Baptists. Unity means that we respect differences of other believers and walk together in our Kingdom work.
I define innerancy similar to the way the Chicago Statement does. If there is a confusion as to what it means, it is because of ignorant confusion as the Chicago Statement and BB Warfield have clearly defined the term.
Saying that innerancy was not really used until 1970's is false. The greatest book written on innerancy was written by B.B. Warfied. Thus, the definition of innerancy was rooted in Warfield's defense in his book Revelation and Inspiration (most consider it one of the greatest early works on the subject). As well, statements like the London Baptist Confession 1689 has one of the strongest statements on the Bible (of course the Westminster Confession as well). The Reformers' call of "Sola Scriptura" as a doctrinal divide demands a belief in innerancy as well. Martin Luther said, "I have learned to hold only the Holy Scripture innerant" (Luther, Martin, What Luther Says: An Anthology). Luther was making a commentary on a text from Augustine and few will doubt that his call for Sola Scriptura included, but not limited to, innerrancy. Thus, there is a line of believing in Innerancy from Augstine, to Luther (and I should add Calvin), to Warfield to today. Of course, most credit Warfield for his tremendous influence on the writers of the Chicago Statement on Biblical Innerancy, including the principle author. Spurgeon even noted that the deteriating view of the Scriptures was one of the first steps in the Downgrade (into liberalism). He noted that liberalism did not occur once someone embraced socinianism or universalism (or, in our day, denied the resurrection or virgin birth). He noted throughout history, the first step is a subtle attack on the Scripture.
In other words, to say that this word was a 1970's invention, is flatly wrong. Actually, it was rediscovered in the 1970's after years of neglect in Baptist circles (though, some of our Presbyterian friends have been holding firm on the Scriptures. Thus, the divide from old/new Princeton, the founding of Westminster, the founding of the OPC, etc). While the word is not always used throughout history, if you read the major doctrinal statements, they embrace the concept.
This was the issue that arose with the publication of the Genesis commentary in the 1960's. He was denying the Bible was true. That is the issue today.
Calvin was tremendously concerned about people making less of the majesty of God's Word. He says:
Let the Pastors boldly dare all things by the word of God.... Let them constrain all the power, glory, and excellence of the world to give place to and to obey the divine majesty of this word. Let them enjoin everyone by it, from the highest to the lowest. Let them edify the body of Christ. Let them devastate Satan's reign. Let them pasture the sheep, kill the wolves, instruct and exhort the rebellious. Let them bind and loose thunder and lightning, if necessary, but let them do all according to the word of God. (Calvin, John. Sermon on the Epistle to the Ephesians) (emphasis mine).
Calvin referred to this as the Majesty... a major theological statement. He later uplifted the Scripture in such a way
We owe to the Scripture the same reverence which we owe to God, because it has proceeded from Him alone, and has nothing of man mixed with it. (Emphasis mine) (Quoted by J.I. Packer, "Calvin the Theologian" p. 162)
This statement drips with II Timothy 3:16, that God's Word is Breathed out by Him and thus significantly Holy and should be revered as such. The people who wish to downgrade the Bible as merely a book that should be given less reverence to God... then innerancy is an issue and should be addressed... Such people are attacking the Word of God that has no mixture of man... but is purely from God. That is significant. Even if no other doctrine was denied, I would deem such on the downgrade, as did Spurgeon, and would disassociate with such a person if I could not help correct their path.
Another note, I know of people in the state convention where I live who deny the virgin birth. This caused an uproar that it helped form the SBCV. In fact, I think the SBCV produced a video about 6-7 years ago interviewing one Pastor who had to fight against his own association over that very issue. However, the isuse is the Bible... the Majestic Word of God.