So, a 2013 998 RT that's overheating.
Obviously, you need a proper level of coolant in the tank and ensure that the front of the radiator isn't covered in bugs. It appears you've done all that.
There is one temperature sensor that feeds the ECM. Then the ECM drives everything else, like the gauge, the fan and the trip into Limp Mode. It appears that your sensor is sending a valid signal.
So, going to your thoughts, if the fan isn't working it can either be the fuse or the relay. (Well, and the fan maybe, but that's another story) And then. of course, we run into BRP's sloppy editing of their reference material. The fuse and relay are in the Left Fuse Box - left as your are sitting on the bike. Attached is the diagram of the Left Fuse Box from the 2013 shop manual. You'll notice that the labelling for the J-Case fuses are backwards from what you have in the back of your Owners Guide. The numbering, 1 - 5, is correct. The labels in the table below the picture are backwards. My sense is that your Owners Guide is the incorrect one, because, the fan fuse is next to the fan relay in all of the other guides. In other words, if you checked the fuse, as indicated by your owners guide, it was the wrong one. The wiring diagram shows that the fan fuse is J-Case fuse number 5, not number 1. And that is consistent with the other similar 998 machines, where the Owner's Guides were corrected.
So, check J-Case fuse number 5 as shown in the attached picture. Don't just visual it, but remove it and either test it or replace it. If all is good, check the Fan Relay. Check that it is all the way in and seated. To remove it, you can grab the thing and pull straight out. Check the pins for corrosion, cleanliness, etc, and re-install. To replace it, you can find equivalent relays at the auto parts store of your choice.
