I don’t bother with a proxy host or with LetsEncrypt, though I guess you could use LetsEncrypt perfectly well. Back when I was doing this, LetsEncrypt didn’t exist and you had to actually pay for public certificates, so using locally generated free ones saved money. It also had a minor(?) security advantage in that if the private server key somehow leaked, it wouldn’t let people impersonate our internet domain.

For the private CA I simply used the crappy CA.pl script that comes with OpenSSL or did at the time. There are much better ways to do it, especially at any kind of scale, but CA.pl sufficed dealing with a few development machines.