You can open it up on the lathe some to hopefully get it to fit. but depending on the shroud size it may not be enough. I have modified one of these tools to make it work on some actions, but it won't fit them all.
Unfortunately I do not have access to a lathe. I think if the " grab arm" ( for lack of a better term) was longer it would work. Maybe I'll go see my fabricator neighbor.
Don't know how the Cooper bolt shroud looks or is removed, but you think it looks enough like a Rem 700's to use the klein. There are field expedient ways to disassemble the Remington bolt, I've used 550 cord and boot laces to name a couple. I bet you could google some videos.
I had trouble with the Kleinendorst fitting the shroud on a number of 700 clones, diameter & length. Found the Brownells unit was slightly larger diameter and because it had the screw knob attached to hook, more length. It's my go-to now and if I remember, a little cheaper.