I believe that you should encourage to participate in the Lord's Supper any and everyone who, if he or she were a member of your church, you would not discipline out. That states my understanding of the extent of the Lord's Supper in its entirety.
A few corollary thoughts:
This presumes that your church has the framework in place to exercise church discipline and the guts to do it.
Our church is a Baptist church. That means that if one of our Sunday School classes started sprinkling infants and refused to stop, they would be subject to church discipline simply because they were sprinkling infants. Believer's baptism is not just our preference, it is the clear and indisputable teaching of God's word. Thus, any pedobaptist member of our church is necessarily someone against whom we would start discipline proceedings.
The reason why I never make statements about the extent of communion using language like "Like Faith and Order" is because too much of a focus on baptism erroneously and dangerously conveys the impression that so long as you are saved and have been dunked subsequently, you need not consider the matter further. But truly every Christian ought to examine his or her own heart and ask the question, "If my fellow brothers and sisters knew about all of the attitudes in my heart and all of the things that I've done this week, and if I persisted in them unrepentantly, would I be a legitimate candidate for church discipline?" If the answer to that question is "Yes," then I need to spend some time getting my heart straight with the Lord before participating in the Lord's Supper. I tell people that only those who are believers and who have repented of their known sin should participate in the Supper. I further clarify that having refused scriptural baptism is a sin.
It surprises me not at all that a sizable number of SBC churches are probably basically Stoddardian in their approach to the Lord's Supper since church discipline is all but lost among us.
In my opinion, it is far more important (and is prerequisite) to recover a meaningful idea of church membership before trying to repair what has happened to our theology of the ordinances. It is difficult to make lasting and meaningful repair to the crack over the doorway before addressing the problems in the foundation.
I am actually optimistic in the long term. More is being written and preached about ecclesiology today than has been the case for at least a couple of generations preceding us. Biblical preaching always bears fruit. I think that this problem will solve itself with time and with the help of the Holy Spirit.