The lens I am looking at is the [Tamron] 18-270 F/3.5-6.3 ... I am fairly certain of full compatibility of this lens with whichever body I end up choosing, but maybe someone can confirm this for me?
Yes, it's compatible.
In addition to that, I am considering purchasing a Tamron SP AF 2x Teleconverter (or other similar if there would be a better combo).
Are there any strong reasons that these would be a bad combo?
Image quality will be softer than using a 600mm lens. Of course, a 600mm lens will be a lot bigger, heavier, and expensive. The more extreme the teleconverter, the more likely there are to be image quality compromises, as with any add-on lens.
Secondly, the nature of a teleconverter is that it increases your lens's focal length. Which reduces its maximum aperture.
Remember, that the aperture f-number is defined as:
focal_length / diameter_of_opening
So, a lens of 100mm with a aperture opening diameter of 50mm is at f/2. When you add a TC, you increase the focal length, but not the size of the aperture opening in the lens. So, in this case, adding a 2x TC would give you a 200mm f/4 lens. A 2x TC adds two stops to your lens's maximum aperture.
A 2x TC would not only make your Tamron 18-270 F/3.5-6.3 into a 36-540mm lens, but also an f/7.1-f/13 lens. And the D5x00 bodies can't autofocus if the maximum aperture of the lens is smaller than f/8 (not enough light is let in by the lens wide open to focus by). This is why the basic recommendation is that 2x tcs should only be used with f/2.8 lens.
You might be able to get by with a non-reporting 3rd party 1.4x TC, which would only bring the max aperture up to f/5-f/9. But chances are good the AF performance of the lens could take a hit if you use it at the far telephoto end. Which most people do, when they get a teleconverter.
See also: "Using Teleconverters" on the nikonusa.com website.