If you seriously have a 3K budget, there is no question that you should go Wacom. Are they expensive? Yes. Are they the best in class? ABSOLUTELY YES. The purchase will easily last you 5-10 years. This is a case of "doing it right the first time" without any regrets knowing you got top of the line. If you're serious about art, drawing, and the tools you use, then I can't recommend anything but a Wacom Cintiq.
For you, from what you're asking, I believe the perfect product would be the Cintiq Pro 16". It's a perfect size, even beyond your 13" recommendation, plus it's a 4K resolution screen, has excellent Adobe RGB configuration, and half your budget, $1500. It's right down the middle and everything you could want.