Mixed. I normally do not bring the issue up in community. I was once asked by a fellow Carmelite, “Why do you want to be a Carmelte, we aren’t Eastern and we do nothing with them.”
Besides the Discalced Carmelite Nuns, I immediately think of
the Russicum which was built with funds raised by Carmelite friars. Although raised for a different purpose, the excess funds were allowed to be used for the building of the Russicum which educated priests for mission work with Russian Greek Catholics. Also, before my time there were Carmelite sisters who sang, from behind a grill, the Divine Liturgy at Our Lady of Fatima Byzantine Catholic Church in SF. They were
invited to SF by the Latin Ordinary to pursue their interest in service to the long suffering Russian people.
So as a Russian Byzantine Catholic in SF I’d say we have certainly been served by the Carmelites, tho not by their clergy.
We had a Dominican serving as a deacon with us up until his ordination to the priesthood, but he was canonically Ukrainian with permission to be a Dominican, bi-ritual to the Latin Rite. There are two young men I know of who are Latin Church Catholics discerning a vocation to the Dominicans who both have some interest, one absolutely has hopes, in becoming bi-ritual priests.
I know Fr. Brendan, Dominican,
Little Boy Lost referred to, from three classes on Holy Icons I’ve had with him. He seems to have deeply embraced the Eastern spiritual liturgical mindset.
Our pastor is an OFM Cap. monk. I don’t think he’s been celebrant of the Roman Rite in decades. When not offering liturgy for us he is a professor at the Dominican School of Philosophy and Theology.
I hope you are in spiritual direction with a priest at the local Byzantine Ruthenian parish you worship with on Sundays. I live in “both worlds” as a catechist in a Latin parish, and frequent daily Mass attender at that parish, or the local Latin Cathedral or the Dominican Priory (that being my personal preferred place to be). It’s a challenge in many ways, including for example being on two different liturgical calendars and two very different praxis. Daily Eucharist for example is antithetical to the Great Fast we are in as Eastern Catholics/Orthodox.
I’m very grateful to the local Latin Diocese which gave me the opportunity to study in a three year non-degree program for preparing master catechists. (They now have their Sacraments/Holy Mysteries class taught by an Eastern Catholic priest who is a wonderful teacher.) If my EC parish closed tomorrow I would still be attending the Russian and Greek Orthodox parishes I go to for services. I could not imagine the grief I would feel if I could not pray as an Eastern Christian.
I did follow the “attacks” over on byzcath a few years back, and some of the guys just got plain old nasty. 
I think the problem most folks had at the time was why not try your vocation with one of the struggling Byzantine monasteries before joining a Latin Order. I totally understand where Br. David is coming from and why he did what he did. 