I've tried powering the project with a 3.7v 18650, a 3x AA battery pack, and a 9v - no go. I really think the power angle is the wrong one.
I have 3 different types of ESP8266 - I have two ESP-12s's, ten ESP-12f's, and one generic ESP8266 by AI Thinker that doesn't have a model number on it (well, it says ESP8266MOD as the model). Here's the thing: both ESP-12s's boot with no issue. All of these are in the exact same layout - same voltage regulators, same power supplies, etc. All of the others fail to boot with the same dim, solid blue LED.
I've just ordered some more 12S's - if the new ones I get in boot, then that solidifies the fact that there is something different about the 12S and the other models that is causing my issue. Other than that, I'm can't think of much else to test.