It should certainly be possible with the gear you describe.
You could set up a script on one (or both, why not?) of your BBBs and have them test dns connectivity periodically, and if it goes down for longer than you consider acceptable change
the router settings away from the Pi-hole.
My suggestion would be to have the script connect to your router by ssh and change the router DNS settings back away from your Pi-hole that way. (If your router supports using ssh keys for authentication all the better).
I'd be wary of having the scripts too eager to change settings like that, so I would also add some sanity checking, such as ensuring connectivity in general is still in place, and that 8.8.8.8 is reliably reachable, and returning sensible numbers before changing back to it.
A less automated option, and also handy to have around as plan C if plan B above fails, but involving some participation by a household member: Have a saved copy of your router configuration but with 8.8.8.8 as the DNS saved on one (or more) of their computers, and show them how to upload that to the router. That way they don't need to worry about what what settings to change, or anything like that. Your instructions for them would be as simple as Click bookmark, type username & password, go to "configuration", click "restore setting", select this file, click yes.