It seems like your regulator is seeing a brownout event. Which one is that, 5V or 3V3?
Maybe you could try using different values for decoupling caps?
Another thought... some time ago i had a lot of problems with Olimex WiFi module. For some reason it caused 3V3 regulator into brownout.
If you can replace it for another WiFi module it would show the fault... or better yet an equivalent Wemos D1 mini module which is a better alternative and easier to play with.