I rode an Allez with nothing more than clip-ons and some race wheels my first year. A tri bike is a definite advantage, but I think at $1000, you'll be wanting something more pretty quickly. I think for your budget, the best option is some used aero wheels. Definitely some aero bars, too if you don't already have them. That way, if you decide later on to get a tri bike, you already have some good wheels for it anyway.

^I don't know, man. I do short course almost exclusively and would never trade in my tri bike for it. The demand on your muscles is so much more intense and you have a much shorter time to adjust from biking to running and have to do it at a much higher pace that I think the tri bike is actually more essential in a short course race. The faster you go, the more important aerodynamics become, which is why a tri bike has such a big advantage for the shorter distances. For example, the difference in drag from 22 mph (good HIM bike average) to 25 mph (good Oly average) is almost a 50% increase (drag is proportional to the cube of velocity).

As far as hills are concerned, when you're standing up, the geometry doesn't really make a difference any more because your only contact points are now the pedals and bars, so you can put yourself in any position you want on both the road bike and tri bike. The road bike will be a little easier to climb on because of the weight, but unless you're doing an all uphill tri, the aero benefits on the flats will far outweigh the weight penalty on the climbs.

In the OP's case where the budget is the limiting factor, the road bike is probably the best option, but when money isn't an issue (if we could all be so lucky :D), I think that a tri bike is always the best option (except maybe Escape from Alcatraz).