How can I keep a momentary button low long enough for microcontroller to read it?
The EXT_RSTB input on the Expressif 8266EX is level-sensitive (low active) so stretching the push-button press will not help- the processor only boots once the EXT_RSTB signal has been removed. 

You can use a couple one-shots to stretch the pushbutton presses (perhaps a 74HC123) but you'll also have to combine the two switches to create one reset signal. If you pull the inputs low, you can use an AND gate (half a 74HC00, for example).  







simulate this circuit &ndash; Schematic created using CircuitLab