Can you hook just he UART cable to the BMS to make sure it's putting out 10-12v? I haven't tried that little switching regulator on anything less than 10v.
if you are, indeed, getting 10-12v on the VCC pin from the BMS, then plug the UART cable into the Wi-Fi module *without the ESP chip plugged into it*. Then insert a pin into the 3.3v connector where the ESP plugs in and one into the GND and make sure you have 3.3v between them. If you don't, then something has happened either with the voltage regulator or somewhere in between. I always test my voltage regulators before soldering them on. Now you may have a different BMS than me that behaves differently but the Engineer @ JBD I talked to said it should work on all of them.
The link John provided might work for you. Personally, I needed 15 of them, so it was a no-go for me to spend that much on each one.
I hate to bring this up, but I saw this just today:
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You'd have to split out the UART cable, but this might work. I have no idea what kind of voltage regulator is on here, so I don't know if it'll work OR even if it will even go from 12v to 3.3v (says "accepts 5v from your arduino," so maybe). I can't really see what the regulator chip is.