Ring, NEST and other WiFi doorbell manufacturers sometimes advertise the Byron 776 as being a compatible product. In fact this is not always the case. 

The Byron 776 contains a built-in transformer with a rated output of 8V 1A AC. Providing the WiFi doorbell can be operated from this voltage then it should be no problem. 

You would connect the WiFi doorbell to the 776 just like you would if you were installing a traditional push button, to terminals 0 & 2 in the 776 and onto the relevant terminals on the WiFi doorbell. 

However most WiFi doorbells operate from a much higher voltage usually 16-24V  and as such the Byron 776 cannot provide enough power. Based on this the 776 would not be suitable. 

There are no tips or tricks with the Byron 776, and in most cases for any issues with operation you would need to contact the support department of the relevant WiFi doorbell manufacturer.