I know it has to be a bit larger, but can I get quality laser print paper to go in a commodity printer? (I have a bizhub C360.)
If so, what size should I get?
You can get any kind of paper to go in it, provided it's the right size. The question is whether it will come out the other side, and what it will look like when it gets there. :-)
Okay, the real answer: A professional-grade printer like the Bizhub (it's a fairly long way from what I'd term a "commodity printer") will handle any paper you throw at it if the paper is rated for laser printing. The paper must be rated for laser printing so that you know it won't scorch, stretch or curl as it goes through the fusion roller. This is important in any case, but it's critical if you plan to print both sides.
If you have the Fiery controller, which is a professional RIP, you have plenty of control. The Fiery understands ICC color profiles, PDF .joboptions and speaks Postscript. It allows you to store the post-RIP document for repeat printing any time, just like any RIP.
I've run 12x18 through a close cousin of your printer, then trimmed to 11x17.
This probably goes without saying, but I've seen great printers destroyed too many times: make sure to use their toner, not an off-brand. Besides the question of color accuracy, it's just better for the printer. In the last few years I've seen four different businesses who complained about their "lousy" (professional graphics level) printers, only because they'd fed them garbage and the poor things died of toxic overload.