The problem with giving it extra water is it needs nourishment. Having belly full with water it cant eat mommas milk. You can try to compromise with having some sugar in this extra water. Its always some calories.... when at the vet can you ask them for a suitable syringe and glucose sugar?
The kitten you can perhaps have at your chest? So it wont be chilled of in the car
The kitten you can perhaps have at your chest? So it wont be chilled of in the car