I posted on the Byzantine Forum for about four years until a year ago, when a strange change in the political climate there caused me to be purged (ironically, for caring about the Catholic Church - pointing out the contradiction of BCs trying to be traditional while the Roman Rite in practice has become Protestant).
To be fair, Byzantine Catholics dogmatically don't deny the Trinity, the existence of hell, etc. There are heretical posters on that forum but they don't represent the doctrines of the Catholic Church. (The Orthodox get misrepresented there too - by an open homosexual, 'Axios', who apparently fibbed his way into an Orthodox church. I'm sure his bishop would be outraged this fellow is claiming the homosexual lifestyle is an Orthodox option!)
Sincere BCs believe there can be only one Church, even though some outside it, including the groups whence they came, have varying degrees of Churchness, and that the Pope of Rome is its designated earthly head.
I am familiar with the types described here.
Seraphim is right that there is the well-meaning BC who ends up coming off as kind of arrogant, putting himself above the magisterium (teaching authority) of his own Catholic Church but also denying the authority of the Orthodox he otherwise emulates. Such might use the jurisdictional rows among some Orthodox as an excuse - strange, since this hypothetical person (I am not anonymously attacking anybody in particular) loves to rub Catholics' noses in it by siding with the Orthodox against them almost all the time, but when cornered as to why he isn't a member of the church he ostensibly believes in, he falls back on an argument the most anti-Orthodox in his church use.
AFAIK Eastern Orthodoxy teaches that the Orthodox communion is the Church and has grace, and that everything outside that communion is a big unknown. Surely such people realize that.
Such, and the not arrogant, more logical BCs who are working to delatinize their churches and criticize liberal mistakes in the Roman Rite, tend not to be born BCs but rather converts, either former Roman Riters or former something else.
Another annoying view among the BC elite online, including that forum, is sheer contempt for traditionalist Romans, their ostensible brother Catholics who share an orthodox, dogmatic worldview, who come to their churches for refuge, an attitude these BCs cop from the liberal powers-that-be in the Roman Rite. I personally never have met a Roman refugee who tried to force his rite's practices on his hosts like this elite claim happens.
The rank-and-file born BCs are pretty much assimilated into Roman Rite ways in the US and don't really want to be Eastern. Definitely true of Ruthenians. (Given the demographics now, they'll probably all be Roman, Protestant and secular in a generation.) Their Ukrainian Catholic cousins do hold onto a separate, semi-Eastern identity but that's out of nationalism. And that construct of Ukrainian identity can make them deliberately un-Eastern in ways too - they'll adopt Polish Latinisms to show they're not Russian, but brandish Russianisms to show they're not Polish. Melkites OTOH aren't very latinized but the French colonial influence caused a change or two.
Russian Catholics, three tiny US churches actually made up of non-Russians, are sincere - it's just that they believe in 'the Pope thing'.
None of the hybridisms one sees in BC practice ever were supposed to happen. (Russian Catholics are proud of the fact they have very few such compromises - strangely, though, their church in New York uses both the Gregorian calendar and the Western date for Easter.) There is a stack of papal documents telling them to stay just like the Orthodox liturgically but often the BCs latinized themselves.
I don't think the essence-vs.-energies distinction (I'll be honest, much of that stuff goes over my head) is seen as a dogmatic difference by Catholics.
