x2^^^ everything you said would be a great start, start off with the big three if your hard for cash and then a better battery and then a high output alternator. i wouldnt use a cap, because its basically a band-aid for your system, if you really want to have one though, get it after everything else.
I am kind of with this guy. I see the route suggested but what are THE BEST batteries to get and how do they help correct this problem? The Audio Shop I deal with swears by an SVR. Supposedly they are also a little cheaper but I know nothing about batteries!
to your list, - the cap, + the alt..I fried my last stock alt i had....and i've been running it on my new one for like a monthand its startin to go bad already. save yourself the trouble and just by a big alt.
better battery and the big 3 done should keep it above 12.5v (you are safe as you are now) unless you are wanting to avoid dimming. svr's are good batteries but you can get kinetik that has more capacity for probably around the same money. with what i see in your sig an hc1800 or larger would work solidly