Well, it seems I've finally figured this one out thanks to analyzing @ILikeBaguette25's run and a bit of practice.

Right after the first grand checkpoint when you meet Sunny and Goderic for the first time for your "reward", there's a small puzzle to solve followed by three clockwork knights to kill. If you can get into the inner circle (see diagram below) before the screen fades, the game will consider you to be past the trigger which normally initiates the aforementioned sequence, and place you in the position immediately after the fight, skipping the entire sequence.

It's not too difficult to hit once you get the hang of it — it just requires charging right before the trigger and then immediately jumping to reach the inner circle, or even slightly past it. The difficulty lies in the fact you only have about a second or less during the fade-out before you can no longer move. The marks are very approximate, but should get you close enough. Loading the checkpoint right after the fight is a good place to practice where best to charge and jump in order to see where you land.

If done successfully, it easily saves around 30 seconds. Good luck!