You mentioned the 3 things that might help (wiring, battery, caps) but you haven't solved the problem. Your atlernator doesn't put out enough power.

Get a bigger/better alternator. Solve the problem, don't mask the symptoms.

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Alpine CDA-9887, TUA-T500HD, CHA-S634, CHA-1204 & KCA-410C
Pioneer TS-A6981, 6 x GM-5100T
Kicker 2 x CVR15,
MBQ RCE-216
PG MX2
Soundgate LanStar1
PAC SWI-ALP