Agreed that it will be a dicey situation to buy a 310 tire and hope it will somehow fit. Be sure your tire shop has a liberal return policy....haha. Most 300's barely fit. I have read about a lot of guys with Metzlers having to shave or sand down the tire ridge to keep them from making contact with the swingarm, even if the rear tire alignment is perfect. Avons are a snug fit, but usually do not pose that problem. The tire you are asking about is 10 mm wider than a 300, which is a hair over 3/8 inch, (3/16 inch on each side). It certainly would not fit on either of the two K9's I've owned.
If you want a fatter back tire, do what Blacktopper did. He had Matt Moore at BDM build him a K9 with a 330 rear tire. Looks like a beer barrel rollin down the street. The build is kinda tricky and pricey, but BDM can do a good job for you, even on an older model.