So you would like to livestream using your camera? The AVermedia Extreme Cap UVC Capture Card [BU110] has you covered. It lets you capture HDMI sources in 1080p60. It supports Mac/Windows/Linux platforms. You can send videos to your mobile device or laptop for streaming. In fact, you can use your favorite streaming software with it.
The ExtremeCap UVC is available in a SDI version too. You can get it here.