looking for some advise on setting up a sports event livestream with local playback
Yes, there will be problems trying to stream the live-image from a dslr. Usually, you would use a multicam-setup for a livestream such as this, including scenes for intermissions, transitions, etc.

Normally, you would use a mixer like the ones from black-magic, here's a link to it. Using it, you would want to feed an sdi signal from the camera into the mixer and send the image from the mixer to a distributer (like a pc or laptop) which will then be able to stream the image.

Unfortunately, you can't just plug a gopro into your pc and press "stream image".