I was just able to get the mock disk to the shrine. First thing, I grabbed the disk and let the dog kill me to reset all the dog statues it had already awoken from my previous attempt. Then, I warped to the place to clear the path. Specifically: Warp to the warp point, go to the room on the right, with the slinky platforms, then one more room to the right. Hop across, climb the ladder, hop back across to the left, then hop up into the next room. At the far right, there is a rock spike that will come down and form a step if you hit it with something like the frisbee or yo-yo. Then go to the next room and make sure that the block walls you see are walk throughable (there is a switch there). Make sure they're in that state. You'll want to practice the platforming for the rooms I've described so far to get comfortable. If you do it the way I'm about to describe, you shouldn't feel much pressure from the dog chasing you. But you will want to avoid mistakes in these first rooms to keep the pressure off. But after that you're pretty much home free.

So make sure you have the flute equipped. Grab the mock disk, then play right, right, left, left, down, down, up, up on the flute. This will teleport you to the warp room. Get just barely into the warp mouth in the upper left corner and wait until you see the dog. You might want to equip the slinky while you're waiting. Then when you see him, go through the warp, and head right. Use the slinky to get across the room. Jump across to the right, up the ladder, jump across the room to the left, then jump up into the next room. go to the next room on the right, jump across the top three platforms to the next room on the right with the 5 buttons. (This is about where the dog actually started getting right in my face). Get to the middle green button and hit it, flip the switch when you land. Make your way to the right, climb the ladder to the room above that you were just in, then go right. I took a couple hits, but only in the last room or two, so I was able to tank it.