My experience is that you usually get what you pay for. There are some old Nikon lenses that are IMO underpriced because manual focus is untrendy. But otherwise, the old adage holds.
For a cheap 500mm lens your best bet is a mirror lens. The Tamron is said to be okay. But the aperture is fixed at F8, so focussing is hard, it is manual focus and image quality is mediocre, due to low contrast. There might be some mirror lenses that give good IQ, but they cost a lot of money. If your aim is simply to record and/or identify a species, then a mirror lens will do.
Regarding teleconverters, they are only as good as the main lens, and usually a zoom + teleconverter is not a good idea. (There are exceptions.) And usually a 2x teleconverter is a bad idea too, in part because the view finder gets very dark. But also because IQ is often poor. However, I have seen images taken with a canon 600mm lens + stacked teleconverters which are incredibly good. Is that in your price range?
Don has been getting good results with a Nikon 400mm F5.6 AIS lens + TC. See the "Lens musings" thread.