In order to use templates on TikTok streams, you don`t have to use OBS or be allowed to stream form PC. There is a link above the widget code window, made special for TikTok:
Open it in a new separate window and copy its address. Then open it in any mobile browser on your phone (better Firefox on Android - it can hide the address bar). Now, with a widget running on your mobile phone, you can start the stream capturing the screen.