Why not? The challenge would be keeping it open/active. If it were open to the public, how are you going to handle anti-Catholic buzzkills without being accused of discrimination?
So make it a private club. Private clubs can get away with anything (including illegal stuff) but how would you attract members? What are the terms of membership? Ages? Dues?
You could also keep the Catholic thing subtle but I’m not sure that is in line with your goals.
I do like the idea so I can picture this place with a awesome mosaic of Our Lady of Guadelupe on the wall (like at the National Shrine), votive candles on the tables (very Catholic but also very
hygge, books in nooks all over, a bulliten board with schedules of area Masses and Catholic events or events that Catholics would support (like blood drives, food drives) maybe even a collection area for canned goods for the food shelf, wool blend socks for the homeless, etc. Have you heard of
pending coffee? Maybe that’s a tradition this shop could have…
Can ya tell I’ve dreamt of a owning a coffee shop with a little bookshop inside?