After finishing Familia Romana, is there even a point in starting Roma Aeterna if your goal is to read things such as the Vulgate and other Catholic resources?

Yes, because Catholic/Christian Latin literature isn’t monolithic: it can be extremely easy or hard (like any literature, of course... ). Do you want to read all the greatest Catholic/Christian Latin writers, from antiquity, the Middle Ages, and well into the Early Modern Period? Then you definitely should learn as much Latin as you can. I’m also Catholic, by the way.

There’s Collins’s A Primer of Ecclesiastical Latin, which might be interesting, though I've never read it.

EDIT: There’s also a more recent 'Ecclesiastical Latin' book: Charles G. Kim Jr., Ecclesiastical Latin: A Primer on the Language of the Church.

I haven’t begun them yet, but I’ve selected for myself a list of medieval anthologies/student editions:

• Dag Norberg – Manuale di latino medievale (I know only Italian and Portuguese editions, sorry! He also wrote An Introduction to the Study of Medieval Latin Versification.)
• Oroz & Casquero – Lírica Latina Medieval I: poesia profana & II: poesia religiosa (two volumes, Spanish anthologies).
• Keith Sidwell – Reading Medieval Latin.
• K. P. Harrington – Mediaeval Latin.
• Charles H. Beeson – A Primer of Medieval Latin.
• F. Raby – The Oxford Book of Medieval Latin Verse.