The L/R outputs going to a separate amp should work out quite well.
A crossover is still needed before the midrange and tweeter speakers. The crossover is used here to block lower frequencies that would overload and burn out those speakers.
If the speaker unit has two sets of terminals that are connected to each other, those terminals are meant only to accept one amplifier channel feed together with a daisy chain to another speaker system. Do not connect two different amplifier channels to those terminal pairs respectively. You will need to rewire the speaker system so one set of terminals goes only to the woofer and the other set of terminals goes only to the crossover.
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