Gonna have to give it to Mulan, with either Pocahontas or Merida in second. Mulan is the only princess with physical combat skills, though Merida is a superior sharpshooter, and Pocahontas has the hunting skills to hide. Real edge is their horses, which give them speed and mobility.
First to fall would be either Snow White or Aurora, lacking any skills tech, or animal guardians. The Princesses with mountable animals would disperse to hunt down any without a guardian.
Ariel would hide in the water, evading Tiana when she rode Louis out in the sea. Ariel can swim faster than Louis but cannot hurt him. Tiana is extremely vulnerable to drowning, and is dragged underwater when Ariel Free Willy's over the gator.
Pocahontas would hide in the woods, playing cat and mouse with Moana, who is resourceful enough to improvise weapons ala Dutch. Pocahontas is more at home in the forest, and scalps Moana.
Jasmine stays put with Raja to protect her. He's more than a match for any physical opponent but loses to Merida on horseback filling him full of arrows from a distance. Jasmine is easy prey.
The other mobile units are Mulan, Belle, Anna and Cinderella in her carriage. Anna and Belle lack offensive capabilities and can only run. Cinderella's carriage is a mobile fortress but similarly lacks offense. Worst of all, it's on a timer. At the last toll of midnight, she'll be left on foot. She is easily picked off.
Circle closes, isolating the fighting to Corona Castle Town. Ariel is forced to leave the water, losing the home field advantage. She is killed.
In an urban environment, Pocahontas takes to the rooftops to stalk. The mounted princesses make for easy targets, and Anna and Belle are eliminated. Mulan, Merida, and Pocahontas are left.
Pocahontas gets the drop on Merida and kills her. She attacks Mulan but her opponent's superior arms, armor, and training win the day.
The victor is Mulan