You really shouldn't need to put that much weight on an Aeropress. The instructions even mention this. Generally, the lighter you press, the lighter you need to press. Starting with a strong push compresses the grounds against the filter, and makes it harder to press.
The other tricks, obviously, are starting with a slightly courser ground (I use "fine", one step above "espresso" on most big grinders), and making sure you stir well so the grounds are scraped off the filter. If you're worried that a courser grind won't give you strong enough coffee (though, really, "fine" is pretty fine), let it sit a little longer. Most fancy coffee sites suggest a full minute steeping, not the 10 seconds that Aeropress recommends.