I can probably control Fully Kiosk using HTTP from SmartThings, but is there a way of doing it from SharpTools?
If you have Remote Admin enabled you can use its features as a simple REST API interface in order to control the device from your local network. The base URL address is always http://ip-address:2323.
The Fully Kiosk REST API is exposed locally on your private network (LAN), so you would need to make the HTTP Request on your LAN.
As you alluded to, using the Web Requestor driver is a good approach on SmartThings.
I also built a Fully Kiosk Driver for SmartThings which leverages the REST API under the hood. I’ll send you a PM for the private beta access (available to premium subscribers).