This sounds like why BK bumped up the minimum flue system height a bit after your Princess was installed. Have you tried adding 3 ft of warm air pipe as a temporary test to see if that reduces spillage?
My owner's manual requires/recommends 12' of flue which is what I have. I have no desire to add 3' of flue because I'm already at the maximum height before roof braces would be required and honestly, it would look stupid having such a tall pipe on a single story home. I think the extra draft strength would likely improve my experience but it's pretty great already. I get no smoke during cold starts.
I'm not sure whether the recent change to 15' was just to avoid spillage during hot reloads or due to an actual design change. Maybe just an attempt to have a gotcha when owners call the tech line to complain of spillage. Similar to how the thermostat dial doesn't have numbers anymore.
Like many on this forum have described, there are some tricks to help minimize the spillage but the BK design is much less friendly for those wanting to swing open the loading door and load without stinking up the place compared to a noncat. It's just a trade off. Worth it for me.