The problem is getting over 200mm will exceed your budget, at least for a new lens (the exception being a 200mm f/2.8L II plus a 2x TC, which is optically not as good, although still likely more than a match for your 18-200mm).

However, if you're willing to buy used, you can likely find 100-400L within your budget - and it's a great lens, although not quite as good optically as the 70-200/4L IS (really, the differences are minor, and that lens at 400mm will be a lot better than a shot at 200mm cropped to the 400mm FoV.