elveax:
It's not the voltage regulator - I just grabbed my 9v and used it to power on a very similar project with the same voltage regulator, no problem.
If you're 50% over the rated voltage, you can't expect it to work reliably. Period.
Power supplies can be tricky. If the ripple is too high, for example, if it strarts oscillating, or if it can't cope with large surges in current draw, digital circuitry will misbehave, even though your multimeter might tell you that the voltage is 3.3V.
elveax:
all the documentation I can find online says that those two modules are more or less identical.
That's because they are.
There are some tolerances, and two of "the same" chips might behave completely differently under certain conditions that are outside of spec.
Fix your power supply problem first, then pull the boot pins to the right level, as shown in the link posted by Wawa. Measure the levels of these pins while holding the reset button to make sure that they are indeed correct.
If it's still not fixed, post a complete schematic of your PCB.
Pieter