There are three propositions relevant here, to which one may assign varying degrees of probability.
The Catholic Church is the true Church (i.e., all Christians ought to be in communion with Rome and Rome will never teach an error such that faithful Christians would have to break communion with Rome in order to remain faithful)
The Catholic Church teaches X, in such a way that if it ceased to teach X it would be admitting that proposition 1 was false.
X is false.
If one ranks the probabilities in the order I listed above–if one is more certain that Catholicism as a whole is true than that X is the permanent, unalterable teaching of the Church or that X is false–then one should remain Catholic. But even if one gives 2 or 3 a higher probability than 1, or 3 a higher probability than 2, one should remain Catholic. The only point at which it would be dishonest to remain Catholic would be if one were more certain of both 2 and 3 than of 1. that would mean that, functionally, one no longer believed 1.
It’s a bit different for someone like myself looking in from the outside, but in terms of whether there is any insuperable obstacle to becoming Catholic the same principle applies.
The “X” that bothers me most is the doctrine that women are incapable of being ordained.
But even then, it is a very close call for me whether 2 or 3 is more probable, and I definitely give 1 a higher probability than either.
This framework gives us six possible attitudes to Catholic doctrine, arranged in three pairs from most orthodox to downright heretical.
123 or 213 would be the normal order for an orthodox, fairly contented Catholic. In both you are relatively confident that the Church is true and that the Church teaches the doctrine in question, over against any questions you may have about the doctrine considered in itself. You might, if left to your own devices, disbelieve X, or on the other hand X may seem obviously true. But as long as you are more certain of 1 than of 2 and of 2 than of 3, it doesn’t make a lot of difference to your peace of mind or your relation to the Church.
The difference between 123 and 213 has to do with the hierarchy of truths. If you hold 213 on a doctrine you are so sure that it’s an intrinsic part of Church teaching that if the Church changed you would no longer believe the Church. Clearly on some things–like the resurrection of Jesus, or the Trinity, or the Incarnation, and probably also the Real Presence and other basic sacramental doctrines, as well as some basic moral doctrines–213 is the proper order. That is to say, some things are higher on the hierarchy of truths than the claims about the Church represented by “1.” But if you put 1 higher than 2 (either 123 or 132), then you are so confident in the Church (or relatively uncertain about whether the Church has permanently committed itself to this particular thing) that you would remain Catholic even if, to your surprise and shock, 2 turned out to be false with regard to X.
Another way of putting it, then, is that the more doctrines with regard to which you hold 123 or 132, the higher up (relatively) you put trust in the Church itself, compared to belief in specific doctrines. In an extreme case, you might say that 123 holds for all doctrines–that you believe even in Jesus only on the basis of the authority of the Church. (Augustine is, of course, often quoted to support this, though I question whether he meant it as strongly as that. And if he did, he was flatly wrong.) I think that’s a very distorted way to hold the hierarchy of truths.
But on the other hand, if you hold 213 for pretty much all doctrines, then you are going to be perpentually nervous and on edge, because if the Church ceases to teach what you think it teaches, then you might lose your faith in the Church altogether. “Traditionalist” Catholics tend to hold 213 for a lot of doctrines, while other “conservative” Catholics (the ones who get accused of “worshiping the Pope”) hold 123 for most doctrines.
On women’s ordination, for instance, I have a traditionalist friend who says that if the Church changed her position on this, he wouldn’t be able to believe in the Church any more. That’s a statement of 213.
132 and 312 are the “faithful dissenter” positions, when applied to something that the Magisterium proposes as true. People who hold one of these two positions on a doctrine are more confident that the Church is true in general than that the Church actually teaches the doctrine in question. Thus, they dissent from a particular official stance not just because they think it’s false but also because they don’t think it is, in fact, the permanent teaching of the Church. The difference between them is that the 132 dissenter would submit to the Church if convinced that X was really a teaching of the Church. The 312 dissenter would not. So 132 has a better claim to be a “faithful dissenter” than 312.
321 and 231, finally, are the positions that really do seem incompatible with being a faithful Catholic. If you are sure that the Church teaches something and that it is false–more sure of both of these things than of the truth of the Church in general–then yes, under those circumstances it does seem dishonest to go on being a Catholic.
The only difference between them would be the possible path back to reconciliation with the Church. For 321, this would be more likely to involve being convinced that the Church really didn’t teach the thing in question. For 231, it would be more likely to involve being convinced that the thing was in fact true. So, for instance, there are 321 dissenters who have what most of us on this forum would consider seriously mistaken views of Church teaching. They don’t actually reject what the Church teaches but what they think it teaches. There are others who may reject things that many on this forum believe, but regarding which there is real debate among Catholics.