You can talk with DragonP I or any other princess normally, you'll only have a chance to get locked into a princess route (if they have one) after a small warning and I think your screen fading grey temporarily.
To recap, what you'll want to do is start WyvernP route, sometime along her route you'll need to repair her collar, which can be done by the tinkerer NPC inside of MouseP kingdom (She's in the west side of the kingdom, hard to miss), then once you repair the collar, slap that sucker inside of the Mystery box, then do a NG+. The quickest way of doing this without cheats is flying (cucco wings or flying progeny like a dragon) to the
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and going north, jumping into the crystal crater inside of the peak which will bring you into the Kingdom Below (bring a mace or another weapon to smash rocks) then enter the
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, look around for a glowing turtle shell, wait inside it for a minute or so, then you'll be given the option to do a NG+
The only way to recruit DogP without tampering with game variables is to enter her route.