Well, as you know PDK is pretty much a hard counter to Sword when it comes out.
From experience, the only instances I've managed to win was:
1. Run aggro, go first and hit face before overflow
2. Constantly control the board and force the opponent to play low cost drops
But still, if Dragon has a perfect ramping hand I'd say it's impossible to win :P
Try running Maisy with a small core of neutral followers. You can fit this into a midrange list without any real problems. When dealing with PDK you need to clear it while re-swinging the board back into your favor so costly removal options won't do it, even spending 3 mana to get rid of it isn't even good enough in some cases do to how much damage it has already done.
The Neutral Package I used was 3 Unica/2 Lyrial/3 Maisy/3 Grimnir/2 Angel of the Word/2 Counsel of Card Knights.