Then you only have to follow the stricter rules for only the public facing certs.
https://github.com/linsomniac/lessencrypt
I've toyed with the idea of adding the ability for the server component to request certs from LetsEncrypt via DNS validation. Acting as a clearing house so that individual internal hosts don't need a DNS secret to get certs. However, we also put IP addresses and localhost on our internal certs, so we'd ahve to stop doing that to be able to get them from LetsEncrypt.
(You say hijacking the HTTP port, but I don't let the ACME client take over 80/443, I make my reverse proxy point the expected path to a folder the ACME client writes to, I'm not asking for a comparison with a setup where the acme client takes over the reverse proxy and edits its configuration by itself, which I don't like)