Oh my, there is a lot to unpack in that.
Okay.
First of all, the Church didn't actually frown on Christians being slaves, or being enslaved, until relatively late. If we're talking "Middle Ages", the first half was a period in which slavery declined uniformly and consistently on a continental scale despite the absence of any particular struggle to accomplish that change on the part of the Church.
If we're considering a Europe with plantations filled with slaves during the medieval period, there's one obvious question that needs to be addressed: Where did all the slaves go in our TL? India, China, and Europe all experienced a marked decline in chattel slavery within a few centuries of each other. Except as a very peripheral economic practice it stayed gone for nearly a millennium in Europe, much longer in the east. The only medieval moral-economic system that didn't go this route was the Dar al Islam, and that may have been a special case - the practice of enslaving war captives was explicitly legitimized in their founding ethics. To be fair to Muhammed, at the time no one - in 7th century Arabia or 5th century Rome - considered a society without slavery possible. And even so, the previously ubiquitous enslavement of debtors vanished from the Middle East, North Africa, and Central Asia.
Graeber makes a fairly compelling argument in Debt: The First 5,000 Years that this dramatic, but mostly-unremarked-upon change was driven by economic factors. Just not any of the economic factors you seem to be asking about here. It wasn't the economics of cost and benefit that (mostly) eliminated chattel slavery in three continental swathes. It was the economics of war and the economics of debt that did it.
If a civilization regulated or systematically condemned usury, it seems to have been almost impossible to maintain a population of locally-sourced slaves. Debt was the primary means of holding people in bondage indefinitely. It was easy to keep a slave a slave for a lifetime; almost impossible to keep a slave family a slave family for more than a generation or two. People end up free, unless strong institutions maintain slavery. This was true within the Muslim world as well. Slave states where debt slavery was a thing faced periodic crises with much or most of the populace becoming unfree; without debt slavery they had the opposite problem. It took reliable imports, and often laws limiting the liberation of slaves, to maintain the institution. (And race helped once invented.) Systemic slavery is hard.
The only solution we saw historically was continuous replenishment via enslaved war captives. In the White Huns TL, a Christian North Africa opens a source to Europe, which begins employing sort-of-janissaries. The source of the slaves needs to be considered. Not that having a source is decisive - Europe had access to the Slavic slave trade, India to many sources, but it wasn't chattel working the land.
In a sense, I suppose I am answering your intended question, in a backhanded way. If it was down to cost-effectiveness and overall production, slavery would have won most comparisons with systems of serfdom. It's just that - for economic reasons - it was not feasible to make use of slavery to begin with.