St Robert Bellarmine echos what St Thomas thought, which is (partially) confirmed by Trent - that someone who has a "true conversion" can obtain justification (i.e. state of grace).  What is a "true conversion" per Trent?  It applies to catechumens, who are learning the Faith, who want to become Catholics

Yes, and it also applies to those who, unknown to us, may have become Catholic for death. I have posted a traditional Catechism, approved by the Roman Congregation for the Propagation for the Faith here before, that has explained this point.

Mother Church prays for dying souls daily because like Her Divine Saviour, She loves us all and wants to save all souls whom it is possible to save. She commands Her Priests and many of Her Faithful children to do the same. There are confraternities dedicated specially to it. So, it is certainly possible some people may be converted near death, and sometimes God has made this known through His Saints at later periods of time; if we have the genuine and true Catholic Spirit, we should rejoice whenever a Soul is saved. And we should always believe and preach that all need to become Catholic before death to be saved. I agree partially with what you said, it is true, for e.g. that the "salvation by implicit faith" issue confused the matter. St. Thomas taught BOD and taught salvation comes only with knowledge and love of Christ, which could, the Angelic Doctor taught, be provided by an Angel or by an interior illumination to a pagan of good will who was sincerely seeking the Truth, Who is Christ.

As long as the teaching of St. Thomas reigned in the Universities, the Church's Mission continued onward spectacularly. St. Francis Xavier, in response to a question posed by some Japanese, answered with the teaching of St. Thomas, that if a pagan sincerely followed the natural law, and desired to know the Truth he needed to know to be saved, God would provide it. But if instead he sinned grievously and repeatedly in other ways and despised the lights he had to come to it, then he would deservedly die in his error and be lost.

Recently, there has been false speculation that non-Christians can be saved; some even openly say that atheists can be saved. All these are terrible errors and Catholics are right to be concerned about them. The best way to solve the problem would be a dogmatic re-affirmation of the Athanasian Creed. EENS means you have to be Catholic before death to be saved. That's all. The Creed says whoever desires to be saved needs above all to hold the Catholic Faith, and then it begins to explain what the Catholic Faith is. That's how the Faith was believed and preached for centuries. I'm not American, but I can give you examples of good missionary Catholic Priests in America who traditionally preached EENS in this way, and also believed in BOD, and had wondrous fruits in their apostolate; making 10s of thousands of converts, usually from Protestantism. Fr. Arnold Damen and Fr. Michael Mueller are a few of them. See this https://olrl.org/apologetics/churchbible.shtml on Fr. Damen. Good Priests like that are necessary, who really love souls with a Father's Heart and a Shepherd's Care, and wish and desire to save the souls entrusted to them. And Good Faithful who support them as well. But saying BOD doesn't exist is not correct. These Priests simply taught all they encountered they had to be Catholic to be saved.

Edit: If following the Dimonds produces good fruits, then why do some who do that call Sainted Doctors of the Church objective heretics, or whatever it is? Honestly, I don't want to fight, and I wish we would discuss these legitimate issues in an irenic manner. But I feel obliged to warn some of you that what you are doing is only one step away from that faithless reprobate, that impious infidel, named Richard Ibranyi, who calls St. Alphonsus "a salvation heretic". That's not Catholic at all. You've lost the Faith if you do what Ibranyi does, and it's the bad fruits sedevacantism and imitating the Dimonds more or less produced in Ibranyi. Please steer clear of such dangerous errors, dear friends, for the sake of your soul that God Loves. Be simple Faithful Traditional Catholics. God bless. 

No, that's not what St Thomas taught, nor did Trent, nor Bellarmine.  They taught, specifically, that you must seek not only Christ but the Church.  "Seeking Christ" is not sufficient, except at the beginning.  How many protestants seek a false Christ, with a false idea of Christianity and who therefore, reject the Church which Christ founded?  Most.
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The Church is Christ's, therefore when you seek the Truth and Christ, you necessarily seek the Church, which is the ONLY WAY to Christ.  This is why the Christ said that water Baptism is necessary for salvation - because it is only through the sacrament that you enter the Church.  It is only through the Church that you can RECEIVE Christ in the Eucharist...which is, let's not forget, also a requirement for salvation (under normal circuмstances).  How many Protestants "seek Christ" in their false churches, but end up rejecting the Holy Eucharist just like the Jєωs in John 6 sought Christ but left Him when He said "I am the Bread of Life"?  Most protestants are damned for this rejection.
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I agree with you that God will provide the Truth to ANYONE who desires it, just like we have evidence that many natives in various places were miraculously visited by saints who taught them the Faith, just like the Apostles worked miracles to find people who would accept the Truth.  But we cannot separate Christ from the Church.  "Seeking Christ" is not sufficient for salvation unless one ends up accepting the Church and all Her teachings.  This is why ONLY those who are FORMAL catechumens were said to be able to receive justification - because they were FORMALLY accepting the Church by humbly taking classes and asking the priest for instructions.  Any other situation outside of a catechumen, has not been confirmed by any saint or Trent, so such situations cannot be put forth as catholic thinking.  They water down church doctrine on the matter.

Didn't Fr. Feeney deny the kind of baptism of desire you describe here as well?