Problem With Dimming Lights

I have two amplifiers on my 2000 Integra.  PPI PCX2400 (RMS 400W X 2, Max 1800W X 1) and Pioneer Premier GM-X1032 (RMS 60W X 4, Max 130W X4).  Neither running full gain.  Maybe half at most.  I have 2 Pioneer Premier PRS12 subs in a sealed enclosure and the four corners each have a 6.5 and tweeter.  Focal Polykevlar up front and Boston Rally in the rear.

I originally had the OEM alternator (80 Amp) which was quite insufficient.  I installed a 150 Amp alternator from Alterstart in Texas.  The thing is that the lights and dash still dim on heavy bass notes but not as severely.  I have a 1 Farad Stinger cap.  Now bear with me on this.  I am trying to educate myself first before I do any more work to my car.

Will I have to run a larger gauge wire from the alternator Positive post to the battery Positive and/or run a larger gauge Ground wire from the battery post to chassis?

I have not wired the cap into the system yet.  From what I understand, a capacitor is NOT a fix from a starved electrical system.  I have a Stinger Deep Cycle battery that has also not been installed yet.  I chose not to install these components until the heart of the charging system is running properly.  I do not want to mask any problems with a large capacitor and NEW battery.

Any info that you can provide will be greatly appreciated.  Preferably facts rather than opinions.  Thanks very much.

The first step you should take is to upgrade the factory wiring such as the negative battery post to the chassis and the Positive post to the alternator.  This would help the eliminate any restrictions that your system has with the transfer of current to and from the battery/alternator.  This should help with the cure.