First, this is the set-up you should use:
- Hunting Grip
- Hollowed Staff
The reasoning for the Hollowed Staff is the increased reload speed will help cut a few seconds off the clock, and the Hunting Grip will result in +50% critical hit damage since you'll have Werewhiskey activate.
Do not pick up the spear yet. Go to the area to your left, and position yourself around the center and kill yourself. The hearty punch will result in most of the the enemies in this area being destroyed, which will save you a fair amount of time. Now you want to jump off the side until were whiskey activates, giving you 100% critical hit chance.
Pick up the spear. Go to the area to the left and activate the switch there. From here, you have a little more choice. I follow the path taken in the video, in which he first goes for the center switch, right most switch, left most switch, the switch near the exit, and then the top switch. However I've seen others go for the top switch first, and save the switch near the exit for last.
Finally, here is a video walk through of it, in case you prefer visual guides. As you will see in the video, it is possible to beat this without using the hearty punch, but it does it make easier: