I agree entirely about EBD being anxious about class -- they were very lower-middle-class, by the sound of things, plus there was the rather murky issue of her father leaving when she was a toddler, and then living very close by, but unmentioned and with no contact, with his 'other family'. Presumably divorce wasn't much more respectable than bigamy in that world!
Joan Baker is everything she fears and dislikes, and wants to set herself apart from, even if her background was more Rosamund Lilley than Joey and Madge's.
I also think she had an anxiety about Catholicism I've seen in other English Catholics who aren't the the longstanding Tudor recusant families and who worried (worry?) about being viewed in terms of poor, immigrant Irish (superstitious, breed like rabbits, controlled by the Pope and their confessors -- all the usual stereotypes).
I think that's partly behind the intense idealisation of the 'child-like' faith of Tyrolean Catholics in the early books, Joey and Madge's ecumenism, the fact that there's no stress at all on major doctrinal differences between Protestantism and Catholicism (we're not told the school has fish on Fridays because of the Catholics, for instance, and there's no mention of Lenten fasting or fasting overnight before going to communion, or going to confession, and one reason Joey and Jack's marriage is skipped must be due to the fact that there's no time for Joey to take instruction between engagement and marriage, so they would have had to marry at a side-altar or in the sacristy, very quietly. as a 'mixed marriage', with no extravaganza of veil and bridesmaids).