Chosen Solution
I had my PS3 controller disassembled so I could clean it. Once I had it cleaned and reassembled, I went to power it on and I could not get it to power on with the Sony button.
I powered it on by plugging it into my PS3. It turned on with four blinking lights, but after ~15 minutes it never paired with the PS3. I disassembled it and clicked the reset button and re-seated the battery, but these didn’t make a difference. Any clue as to what may be causing the issue? This controller was working perfectly before I disassembled it.
The ribbon connector that connects the button membrane to the motherboard wasn’t making contact on my controller. There is actually a little rectangle of black padding that goes between the white plastic shell and the ribbon cable. There is even a groove in the plastic that the pad fits perfectly into. It had fallen out and I didn’t know where it went. But as soon as I placed it back in, the controller powered back on and turned my console on.