In what ways might capitalism clash with Christian ethics?

In every way. Bringing church into state creates conflicts between the beliefs of those who believe and those who do not

Capitalism has a bad habit of wage redistribution which leads to classism. Christianity on the other hand says we are all children of God. Thus class and worth are not determined by man, and Christians are asked to see Christ in everyone. So they can go together, Christianity and Capitalism, but not when capitalism has decreased to an oligarchy capitalism, they no longer line up. However, I am not sure Christian Socialism is actually a thing but other than an Idea. As most of the Socialist countries are anti-religious or multi-religious.
I don't believe that Christianity at its core…  Read more is an economic system. There are several bible verses about Jesus and the money changers, and doing miracles on Sunday. E.G. give into Caesar what is due Caesar give to God what is due God. Christians are to love and server one another. More of an economy of Love, than all of us owning everything like in socialism. God takes care of us, we don't have to take care of ourselves, we have to take care of each other.