I think your budget is the limiting factor. If you're going to get an astro-camera, it should be cooled. And unless you can find a killer deal on a used camera, you're looking above your budget.
That said, I would honestly rather upgrade the D80 with an astro mod than buy a new DSLR that isn't sensitive to Ha. You're missing out on so much date. Here's the difference between a stock 5D IV and my new 2600MC on Carina
http://www.iceinspace.com.au/forum/a...e.php?a=273295