I've actually done this, here's the link:

Here are the rules I used:

1. Timer starts when the front gate closes.
2. Timer finishes when the Juggernaut Lynel is killed (I don't know if there's an offical name for that lynel varient)
3. No glitches (I am using the quadruple break power on the 'Royal Guard Claymore' I don't know if that's a glitch, if it is then this run is invalid)