lagering?

Generally, yes. I've lagered as long as 7 weeks at 34 degrees, and still had plenty of yeast for carbonation, although it was slow.

If you're in doubt because of a long lagering period, and a super clear beer, you can always add a bit of yeast at bottling. What I've done (when I was afraid it wouldn't carbonate) is add about 1/3 package of a clean ale yeast like nottingham to my cooled priming solution. Stir well, and then rack the beer into it.

You don't get any flavor effects from such a small amount of fermentation for carbonation, and it worked very well.

 

I would guess pitching a bit of the slurry would be ok, but I've never done that so I don't have much advice. The cooper's would probably be ok, but I've never used that yeast either! When I did it, I added about 1/3 package of nottingham dry yeast. I would think the Cooper's would give you similar results, but I can't speak from experience.