How to get Caching Plug-ins to work on localhost with HTTPS?

Follow on to comment is too long so done as answer:

As I understand it your test site works as HTTPS unless WPSC is activated?

1. Another (quick to test) hunch:

   Under WPSC “Advanced” settings: check on “Simple” then check the
   “Late Init” (and/or “HTTP Headers”) boxes – save settings. N.B. if
   you deactivate WPSC these settings are lost and will require
   re-checking.

   You mentioned trying another caching plugin – if so try one that
   executes later without mod_rewrites. I wrote the Country Caching
   plugin extensions for both
   WPSC and Comet cache. So I’d suggest trying Comet Cache; it is not as
   sophisticated as WPSC, but as a plugin author, I found it to be
   better designed.

   N.B. If you are testing e.g. an ecommerce site where what is not
   cached is often more important than what is – then both live and
   test should use the same caching plugin and ideally with the same
   caching method settings.

2. Add contents of yourInstalled.cert to WP’s certification authority (CA) list

   No personal knowledge; but a post about a similar WPSC problem on
   nginx
   advises adding the text of your self signed cert to
   “/wp-includes/certificates/ca-bundle.crt”. It will have to be re-applied after every WP update.

3. Pay a web host for a test environment on a separate server.

   It might be possible to create a LetsEncrypt cert for use on a
   “localhost” but this would be far from straight forward.

   If you are running a commercial site then I would have thought it
   would be worth paying ($5? p.m.) to a web host with choice of PHP
   version, MySQL & cpanel with free LetsEncrypt SSL cert install. Buy
   a new domain, or CNAME a sub domain for use in this environment;
   obviously WP will have to be configured for this new domain – but
   your environment can be a (near) exact match to live. Prevent
   indexing via robots.txt, noindex, and maybe basic authentication.
   Disable site when not testing. When not in use switch to maintenance
   mode and disable/redirect away via conf/htaccess.

“might have to disable https on my local environment, but I’d prefer
to keep everything congruent”

If site serves same content under both HTTP and HTTPS, then limiting test to HTTP again should not be a problem. If caching requirements are straight forward e.g. you do not have to worry about what is not being cached on an e-commerce site or you are not caching by visitor country; then disabling caching on test is another option.