I agree with Bongle that using a fans pushing down to increase downforce is a little silly. I don't think it's going to be illegal, however. The super strict readings of <R06> I'm seeing don't seem sensible to me. If you're saying the fan is illegal because it's increasing downforce and thus increasing traction.... Then wouldn't the entire robot be illegal because it's adding additional weight, increasing downforce, and adding traction? The rule specifically says that the intent is for there to be a "low-friction" dynamic. Not a "low-traction/pushing power" dynamic.
Anyways, back to the gyroscope, because I think it's brilliant, and I want someone to try it. I don't know that the appropriate way to use the gyro is as a stiff structure to turn against. It would certainly work, but the dynamics are going to be weird, and I don't know if it's the most efficient use of a gyroscope. Specifically about the dynamics, if you wind up the gyro to a good speed and then try to rotate against it, it's going to generate an overturning moment perpendicular to the axis of rotation of the gyro and the axis your robot is rotating about. Which is going to try to flip your robot front to back, side to side, or something, depending on the orientation of the gyro at any particular moment. (Cool, huh?)
I think the better, if more complicated, use would be as a control moment gyroscope. If you remember the old science trick with a spinny chair and a bicycle wheel, you've got the basic idea. Here's some videos to jog memories just in case:
Cornell Video
Honeywell Video
Using a control moment gyro is a little more complicated, as you'll need to keep track of the absolute angle of it, and try to keep it vertical, but the axis of rotation will always be the same relative to your robot's direction of travel, so the over-turning effects will be more predictable. It should really do a very good job of keeping you pointed in the proper direction. I'll have to do some math to work out just what sizes and speeds we'd be talking about to get a useful effect, but it's definitely something I'm seriously considering.