I have had the opportunity to compare both my working and non-working installations now, and have some good news (I think!) about this issue.
I found that the primary difference was that on the machine that was working, I had an older version of the Enpass extension installed--6.8.6(working) vs 6.9.2(not working). But with 6.8.6, the Enpass icon appears in prompt boxes and clicking in the box shows available passwords for the site.
In the course of investigating the version differences, though, I saw that there is a new (6.9.3) version of the Enpass extension available. So I installed that instead of 6.9.2... and voila, now Enpass is working as expected on both machines.
There are still some oddities in this matter: On NEITHER machine does "Set as default" work in the Enpass extension options. And on both, I have "Use Orion's Keychain" checked in Settings/Passwords, although I'm clearly not actually using Orion's keychain. This is what my Password config looks like in settings for those interested:
For Enpass extension options, I have selected "Show in context menu" and "Show Inline Autofill Popup Menu."
But other than the settings/options menus not aligning with the expected behavior, the extension is now working properly for me. I'd recommend updating to the latest version for anyone else experiencing problems and duplicating my settings, even if it doesn't look like they should work right.