I have two capacitors between and near to the Vcc and Gnd pins.
It looks like your 1000uF cap is going into the perf-board and is not soldered directly to the module. While 1000uF is a lot of capacitance it is not ideal for this application. Larger parts have more inductance and can't deliver the required current instantaneously.
Another concern I have is (I think) there is no regulator at the module. I would put a scope on the circuit. Ground the scope to the metal shield of the ESP module, put the probe on the Vcc pad. These measurements need to be dome this way in order to see what the power at the module is doing.
One thing you might be able to try is using a slightly higher voltage than 3.3 volts. If you are getting dips this might bring it up enough to work. But if it does, this is not a solution, just a temporary band aid.