Doing some further troubleshooting, this seems to be a conflict with my webhost 1and1’s CDN (which is handled by CloudFlare).
Any thoughts on why CloudFlare / CDN might cause redirect issues with this?
I don’t have any issues with your plugin on HTTP, and I don’t have any issues with HTTPS when my CloudFlare CDN is turned on. But for some reason, using all three at once causes a “too many redirects” issue with WordPress.
Aaaaaand I solved my own problem. Hopefully this is useful to you and/or any users of your plugin in the future 🙂
1and1’s CloudFlare CDN must have the SSL setting on “Full SSL” to avoid this issue. The default is “Flexible SSL”, which was causing the issues I was having. Huzzah!