Are there any people who are members of the Roman Catholic Church yet believe in the Calvinist doctrine of predestination?
I think that if they’re Catholics, then by definition, they don’t.
I’m going to stick my neck out and say “no”
There are other views on predestination than the double one of Calvinism. Looking at the writings of St. Augustine of Hippo may be of quite a lot of interest to you, as he had a near-Calvinist understanding of grace and foreordination, yet within a Catholic context.
Jansenism was probably the closest thing to a “Calvinistic Catholicism”, but it was condemned by the Church as a heresy and was eventually pretty much quashed.
Are we talking Catholic by self-identification, or as considered by the Catholic church? Because the Church defines its membership in terms of baptism and other sacraments, not in terms of belief. A Calvinist Catholic would be a misguided or heretical Catholic, but a Catholic nonetheless.
Even if we’re talking self-identification, there are folks who believe all kinds of crazy things but who nonetheless call themselves Catholic. Given that there are self-described Catholics who think the past five popes have all been frauds, I wouldn’t find it too much of a stretch to say that there are probably a few folks out there who believe in Calvinist predestination but who say they’re still Catholic.
St. Ignatius of Loyola’s Spiritual Exercises clearly state a belief in predestination. It says that “No one can save himself without being predestined.” However, as Polycarp hinted, the Calvinist understanding of predestination (“double predestination” i.e. that the reprobate and the saved are both predestined) is slightly different from the Catholic version of predestination.