The shaft diameter will have no impact in this application. A larger diameter shaft means you can apply more torque (if the router can handle it), but this doesn't apply as the slotting cutter uses very little power to remove material due to the cutting length and (generally) increased cutting edges.
It shouldn't have to be said but as with all routing and power tools, make sure you are wearing goggles etc . You don't want one of those carbide tips flying toward your eyes, especially if you are exceeding the SWL ( Safe Working Load) of the cutter.
EDIT: Spelilng

RE-EDIT: I see you already mentioned the H&S with the goggles Chadwick, doesn't hurt to drill it in though (no pun interned!). BTW, are the shafts really made from HSS? I just ran a file over my router bits and they are definitely not HSS, I can't imagine why they would be either.