OK, here's what I got from the OP's posts, for what it's worth - not much since I'm not him, so feel free to disregard until further clarification - just going out on a limb here.
The Trace head (can't be arsed to look for the manual on the internet, assuming it's even there somewhere, sorry) has an active crossover but probably just one power amp: to use it to itself power either the lows or the highs you have to patch one of the xovered outs back to the FX return (or maybe not, as I said I don't know specifically and I was probably busy playing with LEGO bricks when stuff like this first came out); however, I suspect Triad is not doing it, but instead sending both signals to the two sides of the IPR, thus using the Trace only as a pre and crossover. Either way, two amplified signals, one de-feminated (bottom to 250 Hz) the other emasculated (nada lows, just everything else from 250 Hz up), are sent to each of the pair of identical passive cabs, which happen to be 3-way PA speakers, of course with internal crossovers, but, again I suspect, themselves not bi-amp capable, but only provided with one full-range input each (or with a couple each, parallel, for daisychaining). Also, I previously assumed, and the OP didn't correct me, that said passive crossovers are unmolested, or at least he didn't touch them.
[Not different from what they did in the Eighties, with dubious results: two cabs - let's assume they're identical 4x10s, for the sake of argument - fed amplified signals, one is lows, the other highs. Do the cabs happen to have tweeters? Then of course one of them is there just to plug the hole in the baffle, not really doing squat with this configuration; similarly, the woofers of the high cab don't do the heavy lifting.]
Now, if the 250 Hz selected back at the Trace head is lower than the frequency of the woofer-to-midrange part of the internal passive crossover (assuming the woofer is LP'd at all), even the woofer in the "high" cab is indeed working albeit less than the other one, reproducing the signal from 250 to where it taps out).
Still, why does Triad not experience a loss of volume with this setup? and why the perceived improvement in quality?