A mechanism like this does not exist. The specific example you mention - using the Let’s Encrypt logo this way - is not permitted by the trademark policy either.
For some of the reasoning as to why Let’s Encrypt offers no “badge” or “site seal” (which is similar to what you’re describing), see this post.
for your site, which is an independent party that will confirm that the certificate is valid.
The feeling that you have to do this is not a good sign for user security or computer literacy because people’s own browsers are already validating certificates and already have a built-in feature to show the certificate and its validity. They don’t, in fact, need a third party to confirm this when their existing software that they’re using can already do this task. The PKI generally regards certificate validity checking as the responsibility of the user-agent; hopefully users can believe that the decisions made by their user-agents are correct!