Off the wall thought for the cheap strops...whatever they did to treat it is going to be more or less surface only. Maybe try hitting it with a bit of hot water to soften it up, work it by hand, then try some 600 grit paper to get rid of whatever treatment they used on the surface. Let it dry for a few days, give it a slick coat of oil, and see if that doesn't do you better. There might be good leather underneath that finish.

My Illinois strop was at the very least dyed/stained brown. I'll probably never get all that out, but working it over with sandpaper did work away the burnish and give me a better surface to work with. You might be able to make some improvement.