Sharing my experience in case anyone finds this useful, it was pretty much as I expected, I had the taxi driver drop me off in front of door F (closest to the SP check-in desk) where I had to take my dog, crate and luggage, the check-in process took ~20 minutes where all the documents were checked, an extra form had to be filled up and was attached to the door of the crate and a luggage sticker was attached.
From there I had a short walk to odd-sized luggage where I had to wait for someone from Schiphol security to examine the crate, after doing so I could put my dog in the crate and they immediately took the odd size luggage cart to the elevator, this part also took around 20 minutes (mostly waiting for the security agent to show up), altogether I was in the lounge ~50 minutes after arriving to Schiphol. (I took my dog for a long walk before boarding the taxi so didn't see a point in taking him outside again but I was asked at check-in if they should already call up the odd luggage or not yet)