Getting your character is simple and easy. Using the SWEX proxy is the only way to get data out of Summoners War. This is a Com2us limitation, not a SWGT limitation.
SWEX is written by Xzandro. There are few guides below (created by others) that can help you get SWEX working.
Personally, my primary device is using an old iPad connected to SWEX on my desktop. But I do use the Steam client as well with SWEX from time to time.
Please note that if you are a SWGT Guild, Guild+, Personal or Personal+ subscriber, there is a plugin to keep SWGT updated automatically for you. Simply go to the setup menu and pick SWGT setup with SWEX and follow the guide.