Megapixels would describe the body, not the lens, but you are correct - more mp = greater cropping. You can get a decent D800 used for around $600, and a Tamron 150-600 G2 new for around $1200 or used for under $1000. if you are patient and can wait for the right deal to come around - you can come very close to your budget goal. I purchased the the camera and lens used for the attached images used.

I shot nearly 4000 pictures, mostly of birds, with a similar combo (D800 and Sigma Sport 150-600), and though it is not the "ideal" camera for birding, and the lens is "too slow" for some folks, I think the results speak for themselves. I have also shot with the Tamron G2 and find it comparable to the Sigma.

The last two were shot with my lens and camera, but by a friend who was testing the gear out in advance of purchasing it from me.