With your budget, the 300/4 AF-S is a good bet. I have one, and it's a good lens optically, and with the TC14e or TC14e-II converters, it makes a decent 420mm f/5.6 effective, with pretty decent optics. The converter is of course, a high-quality, not-that-cheapo-when-bought-new item. I do not have the 1.7x converter, but have heard it's pretty good.
The real drawback to the 300/4 AF-S is the way it focuses...it's a bit "nervous" you might say. With the 1.4x added, it's even more-nervous. Thom Hogan's review of the lens addresses these issues better than I can in this short space, but know this: the 300/4 AF-S will fail in focusing enough times in a day when used for "action" work that you'll soon come to curse it, and worry about its AF reliability. Not its AF accuracy, but the worry that it will suddenly hunt for focus on a shot you've been working your way into throughout the day, or that two-frames-then-gone kind of shot. I used to use mine for track and field on the D1h and the D2x; the D2x has maybe the best 'action' AF system of any Nikon ever made, a wide-area, amazingly complex system with two modes, 9 of 11 cross-type sensors, covering MOST of the whole frame, and a four-mode focus area/focus grouping style switch, run by the thumb; In most ways, the D2x focuses faster, and better than my D3x, with the 51 point AF system. Anyway...the 300/4 AF-S focuses well...maybe 85% of the time. It's accurate, yes, but it DOES go one "hunts", and with the TC added, it's much worse at that. The 70-200/2.8 VR for example, focuses about 98% reliably, on the same sports. The 300/2.8 AFS-II focuses maybe 99.9% reliably. Over 700 frames at a typical baseball game, and the 300/2.8 on the D3x or D2x will give maybe 3,4 missed focuses. Seriously. On the lowly D70, the 300/2.8 can NAIL one-shot AF acquisition almost every single shot,all day long. It is THAT good! The 300/4...is NOT.
The "real" difference between the 300/2.8 and 300/4 AF-S of the same,identical generation is that the AF system of the bigger lens is just simply MORE-RELIABLE. You never,ever expect it NOT to nail focus, fast. With the f/4, you expect that, each day, it will fail to focus right and cost your chances. At least on moving subjects. This is based on eight years of ownership of both. If you shoot mostly static targets, the 300/4 is fine.