I am not familiar with the sound system in your car. However, I can tell for sure that you can't use the signal from the front door speakers to feed the amp. As you suspect, they are high-pass filtered. I don't know if your headunit has an unadultered (i.e., straight-up, pre- band filtering) output.
What you can do is use separate amps for each band. Personally, I would only bother with an amp for the midrange and one for the bass. Why amplify the tweeters/trebble? Then again, I am not a big music/soud head as many people are.
Once the energy in any given frequency band has been filtered out, you cant really make it reappear. That is why you need to amplify them separately.
Anyone feell free to chime in. My contribution here comes from my academic knowledge of electronics, signal processing, filtering, and communications. You all may have a street edge that I lack.
Dan, good luck with everything, and welcome to UB.c.
- Dan (yes, me too