It seems to me what Feeneyites are really fighting is Pelagianism.
I think this BoD debate is kind of a distraction. It's a highly technical theological debate that misses the point. The point is that modern men think that heaven is a retirement that everyone is entitled to by nature, and that you can only lose this retirement by being an egregiously evil person.
In order to combat this Pelagianism, we don't need to go down the route of stressing the need of membership in the Church for salvation (EENS), and even going so far as to deny that one must be a full and visible member, which propels you into a debate involving statements from centuries of theologians and magisterial docuмents. I think every time you try to fight this Pelagianism by going down the EENS route, the modernists smile because they have centuries of texts that create enough ambiguity for them to create the impression that everyone in the world is in the Church by implicit desire, by being "a good person" (and 99% of people are good people) you become a Catholic in the state of grace.
Avoid this.
Instead of contradicting all those doctors and saints and theologians that mention BoD, use the very SAME doctors who talk about the fewness of the saved. This is the doctrine you should look to, not EENS. Look at St. Thomas for example, who asserts BoD, but who also says that the number of the saved is few. Most of the great theologians of the Church have asserted that the number of the saved is few, even those that promote BoD.
All you need to do is point out that the doctors of the Church say that few Catholics are saved, and then say, "and if few Catholics are saved, who have all the grace of the sacraments and the 'fullness of the truth', then what are the chances that non-Catholics, who do not have the truth or the sacraments, will be saved? As St. Peter says, "And if the just man shall scarcely be saved, where shall the ungodly and the sinner appear?" (1 Peter 4:18)"
You don't need to point out to Protestants that it is necessary for salvation to belong to the Church, because they can dig up all those texts, and can point to the ecuмenism of the popes, to contradict you. All you need to do is quote Catholic sources on the efficacy of the sacraments (particularly Penance and the Eucharist), and say, "how do you expect to live with you do not have the body and blood of Christ, which Christ himself says, "unless you eat the flesh of the Son of Man and drink his blood, you cannot have eternal life within you", and how can you be sure of salvation when you have sins on your souls and no means to confess them with any surety of absolution?"
I tell you, the modern Pelagians smile at the EENS/BoD debate because they can point to the Feeney scarecrow, "Feeney was condemned", "you are a heretic". That the number of the saved is very few, a doctrine taught by so many of the great theologians from the earliest times, supported by Christ saying that few find eternal life, is not something that they can smile at. At once you have pointed out that few achieve eternal life, then one becomes more interested in the reality of the Church, who has the true doctrine and sacraments.
You know that BoD is their preferred mechanism for justifying their Pelagian attitude that the majority of mankind is saved by being nice and polite, even though the doctors of the Church never used BoD in this way. You know that this mechanism began with the Jesuits a few centuries ago, with the finding of the Americas and the Pelagian need to somehow make these pagans Catholics by desire, "because how can so many be damned? Is it there fault they haven't heard the gospel?" Ignore BoD, ignore EENS, focus on the Ark of Noah / Church parallel that so many doctors pointed to, showing the fewness of the saved.
The problem is that Pelagians imagine that men are naturally good and deserve to go to heaven. They have no sense that man is corrupt, falls very far short of the purpose he was created for, and that even the perfect man has no natural right to heaven, because to be supernaturally joined to God is absolutely above man's nature and can only be a gratuitous gift given by God. They do not understand the difference between good and evil. They want to make heaven appear less wonderful and less gratuitously bestowed upon us than it is, and they want hell to appear less terrifying and less justly given to us.
You don't need to stress EENS to get them to wake up concerning salvation. The fewness of the saved is a more efficacious doctrine to that end (see: St. Leonard of Port Maurice's sermon).