Mac won't turn back on after it dies from low battery

My Macbook Pro 13” Mid 2012 original battery works great (except for the 1 flaw expanded upon below). System information shows the Health condition as “Normal” — I would give the specific cycle count and capacity information, but my computer is currently dead.

HERE IS MY ISSUE:

If my computer falls to 1% and automatically goes to sleep, I am unable to turn it back on. When I plug in the MagSafe charger, the light stays bright green and never changes to Amber. If I plug in before the shutdown, it immediately goes to amber and has no issues charging to 100%

I have opened the back panel to disconnect the battery, drained the caps, and reconnected the battery. This fixed the issue the first 2 times I tried it but it no longer works.

I have also tried the SMC reset more than once.

The most consistent solution seems to be leaving the MagSafe adapter unplugged for 16-24 hours with the battery still attached. After waiting a day and connecting the charger my computer will turn on and the MagSafe will go orange ~75% of the time. The other 25% of the time the computer won’t turn on and the adapter just stays green.

Not sure if this makes a difference, but when I press the small round battery indicator button, one of the indicator lights flashes green about 10 times. The flashing occurs in the same manner whether or not the MagSafe adapter is attached.

P.S. — Please don’t respond with “well don’t ever let your computer fall below 1%”. Obviously I try to do this, but I don’t sit at a desk all day and I’m not perfect.

BOTTOM LINE: I am not in a position where I can wait a full day to get my computer back on. Is there any way to resolve this? Do I need a new battery (seems silly since everything else about it works great). Is there a different component that needs to be replaced?