After looking for a sat nav mount last year to share my BMW Nav V with my GS, I bought a BikePenR mount. OK - try to get over the company name
Euro 140 I iirc, something like £65, shipped
For the K1300S you need the GT10, this mount is available in silver and black. I got silver. It looks good.
Installation is with allen keys and simple.
The mounts are isolated and I have seen no vibration.
There is slightly restricted access to the key - it doesnt trouble me at all. But other than that its a good location. Better would be above the instruments like the GS but its ok.
You also need a cradle - I bough the four button BMW cradle from BMW. It never stops
Overall: happy.
Installation -
Instructions and contact details -
https://drive.google.com/open?id=0B8KUQN1UWNyneThzUUNpZk9vakk