Blessed Theodore Romzha and Bishop Basil Hopko come immediately to mind. Both lived under Communist rule. There’s also Met. Andrew Sheptytsky, whose cause I believe is open right now.
Blessed Paul Gojdich. He asked God that 1) he could receive Holy Communion before he died and 2) that he would die on his birthday. God heard him because that’s what happened. He died on his birthday after receiving Holy Communion one last time.
I’m not exactly sure what you mean when you say Byzantine Catholic, a lot of times that’s used as shorthand for Ruthenian Byzantine in the US, but assuming you mean Byzantine Rite, so like Ruthenian Byzantine Catholic, Melkite Greek Catholic, Ukrainian Greek Catholic, there are more than a few Martyrs that qualify. Especially under communism.
Silly me. I forgot to mention; Ven. Archbishop Fulton J. Sheen was bi-ritual with faculties in the Byzantine (I believe Melkite) tradition. Does that count?
DISCLAIMER: Catholic Answers has turned over the archive to Catholic-Questions.org and no longer owns, manages, or moderates the forums. For additional apologetics resources please visit www.catholic.com.