The separate crossover isn't needed - as you have noted, you can always rely on the crossovers built into the speaker. However, active bi-amping (which involves eliminating crossovers at the speaker and placing them upstream of the amps) is generally considered to be a more beneficial form of bi-amping. Passive bi-amping (the arrangement that you describe in your post, with the speaker's internal crossovers still being used) basically only allows you to feed the speaker more power.

I can think of very few speakers that would be likely to benefit from a change in power output from 300W (at 4ohms) to 600W (also at 4ohms), which is why I initially suggested that this arrangement might not yield a detectable benefit. However, I don't know what speakers you have, so I could be wrong. Could I ask what speakers you are using?