No. Reflector 3 is designed to screen mirror/cast an iOS or Android device to your Mac or Windows PC. Reflector 3 can only receive AirPlay and Cast connections, but cannot send them to AirPlay/Cast receivers such as Apple TV and Google Chromecast devices. These are limitations that Reflector 3 is bound to.
There is a way to accomplish this by using the combination of Reflector 3 and AirParrot 2.