You’re welcome to try to include a Let’s Encrypt certificate in a PDF document, however its key usages are limited only to securing TLS connections (1.3.6.1.5.5.7.3.1 & 1.3.6.1.5.5.7.3.2)
Any real PDF reader would see that the certificate policy is not suitable for document signing and would not considered the document signed.
Edit: I’m not sure what the standard is for verifying signatures in something like Adobe - all I can find is that certificates have to be manually trusted. If you can try use the cert.pem output of e.g. Certbot along with the private key privkey.pem to sign a document and report back, that could be illuminating.