I left the stock wiring, so the HU is wired to the factory tweeters, but I also turned the internal amplifier off. If I switch the HU's internal amp back on, it powers them using the same settings as the door tweeters with how it's wired.

I tried them on vs. off, and I prefer the sound without them by a pretty large margin with this stereo system. The added volume is certainly unnecessary, and I have time alignment and frequency compensation on the main woofers and tweeters. If I have the same speaker output running to a different set of tweeters in a different location, I undo a lot of the time alignment and freqency adjustment on the high end of the range.