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You can use a leather strop, no problem. Before you run into problems, check this wiki article on how to apply pastes to strops.
Second, this article will learn you all you didn't know you wanted to...

It sounds like you know how to use strops, so getting a second strop set up for paste or CrOx sounds like a great idea. I would not add paste to the daily strop, but an old one works great for paste....