After asking around, it seems that there is no good way to fully control the configuration of a WAB website down to the individual widget settings, other than manually. 

We ideally need an api that exposes all the various settings for WAB widgets, (at least the out of the box ones, & would be great if external widget developers could also implement it in their widgets) so they can be controlled via python (preferably) scripts. In an environment where there are many staff working on various things, and many WAB websites for different parts of the client base that are 'supposed' to be configured the same way - it's hard to keep Dev, Staging & Prod all in sync, and prevent any one-off changes from a standard baseline.  We're in the beginning stages of trying to implement a CI/CD process for controlling all our web apps and servers via scripts, but so far WAB seems to require human intervention via a GUI for most everything. Having a way to periodically run some reconfig scripts across each WAB website to ensure everything remains consistent would be ideal.