Campbell's thesis is that Paul wrote Romans to combat the impending or already actual visit of a Jewish false teacher. Most likely this is the same teacher that Paul opposed when he wrote Galatians (probably not that long prior to Romans). Romans is in a sense developing the same themes as Galatians at points. However, the argument is much more developed and even at times slightly different than Galatians. Campbell attributes this to Paul having learned from his experience in Galatia and knowing his opponent better. Romans represents a later round in the debate (see esp. 506-8).
This explains other features of the letter well, especially why Paul was so concerned in the letter with Jewish matters (particularly his espousal of a law free gospel in Romans 1-4). These matters get little of his attention elsewhere, where false teachers aren't on or soon to be on the prowl. Particularly, the question in Galatians seems to be centered around, 'who are the people of God?' Romans 9-11 is a response to concerns over Paul's position, especially with his concern that the Gentiles not become arrogant towards non-believing Jews. Paul is probably responding to critique (509).
This understanding of the purpose of Romans also explains what is probably the hardest nut to crack in the letter - Paul's stance of non-interference - in a letter where it seems that he's interfering. Paul in fact is doing no such thing. He believes the Romans to have been already established in a gospel compatible with his. Thus he's not interfering, merely strengthening them in what they already know (501-3). It's actually a veiled shot at the Teacher. He's the one interfering, both in churches Paul founded and potentially in Rome as well.
Lastly, I want to address Paul's exhortations to the weak and strong over matters of diet. The first thing to notice is that he's not hard on them, and also the discussion is much briefer than similar dietary discussions in 1 Corinthians. Why? It's not that big of an issue in Rome (but it still is an issue). Additionally, it's important to observe that Paul doesn't proceed in this section as many today might expect - 'don't judge those who don't observe the dietary laws because you are justified by faith and not by works.' Paul does not ground this exhortation in his earlier argument. That means that something other than concerns to see the Romans live out justification by faith is behind Paul's choice to address this issue. I think Campbell is correct in believing that Paul brings it up because he expects the Teacher to arrive soon, if he hasn't already. The Romans will need to be as unified as possible if they're going to survive the Teacher (this also explains how Paul's exhortations there tie with other pieces of chapters 12-14 that focus on unity). (510-11).
Overall, I find Campbell's argument to be persuasive. None other that I know can better explain the features mentioned above (Campbell discuss the shortcomings of other positions at length 469-95). The next chapter begins the analysis of Romans, tackling 1:18-3:20. We will attend to that next, but it may take a few weeks as the section is lengthy and intricate.