I think your battery has outlived its usefulness...or you don't drive the car far enough.
Your rear bulb could be bulb, socket, needing of dealer free retrofit, or you have a wire out in the elephant trunk.
Look under your battery clamp for the date on the battery...if 4-5 years old, consider a new 49DL from Autozone. Maybe just start by charging it or driving it further. You need about 20 mins. on highway to build up the battery. If you do a lot of short trips, you might not recover the power you lost by starting it, which is the largest draw of current.
You need not only voltage, but amperage. If you're starting okay, it sounds like you have the amperage, though ironically, I wonder if maybe you don't have the voltage? I don't understand batteries too much...so apologies if that's wrong!
Use cluster test to check voltage when car is off and on...use cluster test to test the cluster too (yes, I meant that if it sounded weird!).
In NJ--Tierney's in Montclair, Sept. 25, 7:30 p.m.
In NYC Sat, Sept. 30, 8 p.m. at Broadway Comedy Club! Sat, Oct. 8, 6 p.m. at Greenwich Village Comedy Club.
Comedy can be a great cheap date!
Mangofy your E46!