Bike is new so haven't done it yet but did see how it is positioned and it looks to be an absolute pain in the ass.
Anyway I had a similar problem on the better half's old R6 when I did brake fluid replacement. What I did then was remove it from its mounting and pull it out from the subframe. From what I can see with the Suzook, this might work as well by removing the plastic side panel under the tank and the right side rear panel. This would allow access to the bolt that holds the reservoir to the subframe. It looks like there is enough slack and flexibility in the brake line to pull it out. You can securely fasten it to the frame at a level position to remove the cap to pour in new fluid.
This is just a suggestion based on experience. Gaining access from the top will be difficult, I think. Either way, I'd bet that as I type this, some guy from Nissin and another guy from Suzuki are drinking sake and laughing their asses off about how funny they thought it would be to position the reservoir where it is.