[rant]i've held off a little because at first it sounds like a "tech support explaining 'obviously technologically complex things' in a 'simple manner' so they can move on to the next simpleminded idiot" thing.
personally, i'm extremely surprised that this thread is the first time i've heard of factory radio's not working with aftermarket gear. i know there's a few vehicles that use canbus, or optical audio connections with specialized protocols, but have never heard of analog systems with this problem before. i honestly think it's something to do with the way audiocontrol integrates/expects the signal to work, but i'm not familiar enough with audiocontrol gear to have any confidence in that. which is why i never said anything before.
with this vehicle, i know there's been plenty of people that have replaced the factory speakers with aftermarket, as well as a number of people that simply added a sub in the trunk with variations of the same method you've used(usually with some more wiring destruction, but connecting to the same wires). never have they presented these issues.
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firstly, have you tried the amp without the LGD? i'm not entirely convinced that they're required. i believe the factory door speaker amp is class D, but i suspect that the radio output itself is more of a standard class a/b.
but what i dislike the most about these modules is that audiocontrol appears to be the only company selling anything like it, not even alibaba resellers have them. you can't tell me that one single company somehow knows something the entire car audio industry and chinese reproduction electronics industry doesn't. i'm not denying that they might have a positive effect, but this module screams money-grab, just the same as those
$60 wooden blocks in high end stereo magazines to "lift speaker cables off the floor and reduce resonance"...
second, something that could be tried for minimal extra cost is to use a Line Output Converter(LOC). this one's $10, could probably find one locally for under $30 at any audio shop, they commonly stock them for the 'add subs' crowd.
https://www.amazon.com/PAC-SNI-35-Va.../dp/B001EAWS3W
a LOC is what would be considered the more 'antique' option to connecting an amp to a factory system. the reasoning here is that it would change the factory wiring 'speaker-level' signal to the more aftermarket-industry-standard of low level RCA jack type signal. it also removes the line level conversion to a different, potentially more adjustable/expendable external module that can more easily be tested/diagnosed.
the benefit to this method is that any aftermarket amp would then work. it does defeat the purpose of the specialty audiocontrol amp(same as installing an aftermarket head unit), but then allows one to test and verify the entire system to extremely well-established methods. literally any youtube video on 'connecting/setting up an amp' will then apply, and you can forego any of the specialty audiocontrol stuff in setup and wiring.
i believe audiocontrol, and other more competition-level ears would argue that the 'cheap LOC' method could introduce some noise, or distort the signal that could prevent getting the most out of the amp/system(keep in mind that this mindset will be specifically geared towards winning 1st place at audio competition events, in a stationary vehicle, using test equipment), but at this point, anything working is better than nothing working. noise can be mitigated with different gear if the result proves successful in making things work. LOC's have been on the market since the market began.
but first, try it without the voodoo LGD's.