Your esp will probably “just” brown out when initalizing with wifi 
MORE 
State Consumption Power down 200 µA Active, WiFi OFF 34 mA WiFi ON 67 mA + 435 mA spikes
Guess the wifi scan/connect/transmit will cause to much power draw. Might work around it with insane amounts of capacitors or by placing a poor man’s UPS (power bank that can be charged and discharged at the same time).
Another idea is to disable wifi completely (just recently introduced in esphome if I remember) and try something more power conservative to transmit the data necessary like esp-now 