Thanks for your detailed help. I took fletch's advice and reinstalled a fresh new version of Perl compiled against my current OS. As far as I can tell, the module compiles. However, it fails its tests. (Please see my comment above.)
My instinct tells me that the module is having trouble finding Augues. I know Augeas is installed and works on my system because I can do this little tutorial found here: https://augeas.net/tour.html
I tried taking a look at the source code of the module to see if I could spot the problem. But all I found was this call to an init function and I'm not sure how it works. I couldn't find a sub called "init" anywhere else in the distribution. Maybe it's an XS thing?
Any other ideas on what I can try to get this working?