It's hard to see what Paul is actually getting at in 1 Cor 11:18-20. For review:
18 For, in the first place, when you come together as a church, I hear that divisions exist among you; and in part I believe it.
19 For there must also be factions among you, so that those who are approved may become evident among you.
20 Therefore when you meet together, it is not to eat the Lord's Supper (NASB)
In v. 18, it appears Paul is scandalized by the report of divisions, even shocked. But then he turns it around in v. 19 and seems to say that it is expected, even natural, and that some good may come of it. Verse 20 verifies that the divisions are really not a good thing.
A rough analogy might be when a mother comes home to find her child has wrecked his bedroom. At first the mother is aghast, but then says, "Well, I guess when you clean it up you might find the ball you lost last week."
Here are some other translations of v. 19:
No doubt there have to be differences among you to show which of you have God's approval. (NIV)
For doubtless there have to be factions or parties among you in order that they who are genuine and of approved fitness may become evident and plainly recognized among you. (Amplified)
No doubt you need to take sides in order to show which of you God agrees with! (NIV Readers)
The best that can be said for it is that the testing process will bring truth into the open and confirm it. (Message)
I like the NIV Readers somewhat, and The Message a lot here. (They have "taken sides" on what this verse means.) I especially like the tone of The Message. I think it may capture the meaning the best.
Unfortunately, Paul doesn't tell us how we will know who is approved. Will it be those who are the nicest? Those who always go along with the official elders? Those who stay loyal to the Ministry. He doesn't say, he just says we will know. It's probably some combination of a lot of things. Observed godliness, adherence to truth, peace-seeking, etc.
But here's an important thing. If Paul were endorsing the LSM brand of approvedness, he simply would have said, Always submit to the ministry. He wouldn't have said that there is a dynamic vetting process of which we can't at the beginning know the outcome. Paul doesn't know who is going to be approved! He just knows that it's possible that someone will come out of a divisive disagreement approved, which implies that someone will come out disapproved.
When you think about it, there is really no other way, other than arbitration, which is--you guessed it--the LSM way. Arbitration by them, or by "the Apostle." That is, the poobah or several poobahs come in and tell everyone who's who and what's what. Anyone who doesn't fall in line is kicked out and blacklisted. But that's not Paul's way in 1 Corinthians.