Does Romans 14-15 help us much when we have a dispute in the church?
We all agree that Romans 14-15 is primarily concerned with a question of eating which is dividing the Roman church. (Secondarily, there seems to have been an issue about observing festival days (14:5-6) and drinking wine may also have been a problem (14.21)
In my experience, perhaps in yours also, we do not seem much agreed about applying R 14-15 by analogy to other issues troubling us these days.
Perhaps we can, perhaps we should not. You may have thoughts on that in comments below.
We might usefully observe, however, verse 3, which reminds us that our unity is in the God who welcomes us: "... for God has welcomed them" where "them" equals "that lot over there with whom you disagree so strongly."
Yet it would be odd, would it not, if we read Romans today for its universal theology of salvation (i.e. its timeless, applies everywhere and to everyone message of the gospel of the saving power of Jesus Christ) yet not for its applicability to the church of today in respect of our disputations?
Having said that, I wonder how you find Romans 14-15 as a "dispute settling" method? Even in respect of eating in Rome, is it clear by 15:6 how that dispute was settled? (And, if it looks like Paul was pushing for the dispute to be settled in favour of "the weak", in the long run, as Christianity parted ways from Judaism, "the strong" won and not "the weak.")
Sure, Paul generally sets out an excellent case for "going along to get along" in Christian fellowship (e.g. 14:19: "Let us then pursue what makes for peace and for mutual upbuilding."). Also, sure, Paul is very clear about not causing a brother or sister to stumble (14:20-21). And, very surely, Paul clearly warns against being "the ruin of one for whom Christ died" (14:15).
But much less surely can we say that Paul is simply saying that the "weak" on an issue always have things work out their way because the "strong" on the issue should always make life easy for them. For example, Paul challenges "the weak": "and those who abstain must not pass judgement on those who eat" (14:3; see also verse 10) and generally urges all sides of issues to recognise the other side as they honour the Lord (14:5-6).
Further, Paul is focused on these matters at a simple level: two groups, one should give way to the other (even as both groups should love, accept and honour the other). He does not get into the complexity of (say) one group holding the other group to emotional ransom; or of one group playing cute political games with the other.
There is also the complexity of determining who on issues outside of food, drink and festival days in ancient Rome are "the strong" and "the weak". Do these neatly map onto "conservatives" and "liberals" in 21st century Anglicanism? Probably not! Do these neatly map in synodical contexts onto "the majority" and "the minority"? Possibly so. (What if the majority in (say) the Synod of the Diocese of Christchurch is part of a minority within the General Synod?)
But complexity should not dissuade us from applying Romans 14-15. There is much in these chapters which steers us to Christ, which reminds us of Christ's teaching (e.g. not judging one another), and which challenges us to be like Christ:
"Welcome one another, therefore, just as Christ has welcomed you, for the glory of God." (15:7)