The large startup current pulse of the WiFi (250mA+)
even when there is a 1000uF capacitor across the ESP32 supply rails.
How about using two regulators? One for the ESP32 and one for the wifi circuit?
The ESP32 has the WiFi built in, you cannot power the WiFi separately.
I haven't used this device, but a LP5912 from TI might work (500 mA, QI is 30 uA).
Alternatively, you can use a boost converter?
Did you try more than 1000uF on the power rail, and were they low-ESR tantalum.Leo..
No, but I did try a 470uF low ESR electrolytic, that did not cure the problem with the 250mA regulator, although with the the 500mA was OK. I dont have a 1000uF + tant, and they are non-cheap.
The resistor you need is 10 -20 ohm, calculate voltage drop.Without schematic - it look like you discharging 1000uF capacitor by voltage regulator that why you burned it, increasing value of capacitor make situation worse.
You need smaller capacitors in parallel with your electrolytic caps. Even low-ESR electrolytics are too slow to be used without tantalum or ceramic caps.