Incidentally, the issue with Bishops in the House of Lords is that they constitute a block vote. Which, when it comes to issues like, for example, assisted dying, stem cell research or abortion legislation is likely to skew the vote towards the traditional Christian position.
This is where the whole discussion becomes unhinged.
Firstly,the Lords Spiritual are 25 out of 771 voting Lords. They have not been the swing vote (ie, if you reverse all their votes, the outcome changes) for more than a handful of pieces of legislation in the past two centuries: the last time they had significant influence was the bizarre and long running trench warfare in parliament throughout the late 19th century over whether a man could marry his late wife's sister.
en.wikipedia.org/wiki/Deceased_Wife%27s_Sister%27s_Marriage_Act_1907
These days usually only one or two vote. If they did, by some miracle, form a swing bloc on something that mattered, then (a) the Parliament Act would be invoked and (b) there would be a constitution crisis which would probably resolve the issue against the Lords Spiritual, which is why they'd never try it.
It's worth noting that the last time they did overturn a government position, it was to ensure that children of asylum seekers were educated in schools rather than in detention centres, an amendment moved by the Bishop of Portsmouth and won by one vote against the government. That's a good thing, right?
Oh, and they don't form a block vote either. Unusually, a load of them turned out for the measure on balloting for the abolition of grammar schools, and split 9:2 (in favour of balloting, ie the non-conservative stance), and for the civil partnership bill, where they split 8:2 IN FAVOUR OF CIVIL PARTNERSHIPS. So evidence for them being a block is non-existent, and for them being a socially conservative block less than zero.
Secondly, at the moment CofE bishops are ex officio members of the Lords. But if they ceased to be ex officio, ex nim says, they would be appointed under their own steam, just as Jonathan Sacks and Donald Soper were.
Thirdly, there's no evidence of their alleged influence anyway. Welby had a pop at same-sex marriage but gave up immediately, and the legislation passed in a landslide in the Lords. The assisted dying debate has never got anywhere near legislation with a chance of passing Parliament, and was defeated in the Commons a few weeks ago 330-118. That's elected MPs voting it down, nothing to do with "Block Votes". It was the Church of England who campaigned for the 1967 Sexual Offences Act which decriminalised homosexuality, and the archbishops of Canterbury and York both voted IN FAVOUR: the opposition to the bill's passage came from backwoodsmen.
I don't know much about the passage of the 1967 Abortion Act, but I don't think that the Lords Spiritual were a major force: that legislation came from the Commons (it was David Steele's act) and there was sufficient support for it that the Parliament Act would have been invoked anyway.
We have legislation permitting stem cell research. We have some of the most liberal abortion law in Europe (Northern Ireland aside, but that's nothing to do with the Lords) and the changes that have - regrettably - been made to it came from the Commons. Assisted dying is routinely voted down in the Commons and therefore what the Lords think is irrelevant.
Catholic Priests can't serve in the Lords for reasons of their own Canon Law (they cannot give obedience to any power other than the Holy See) so they can't take seats even if offered: you appear to be confusing the social positions of the Catholic Church with that of the CofE.