I think ZeroSSL is the right product for you.
For the benefit of everyone, I should probably clarify how certificate installations work with cPanel.
With a traditional (non-ACME) CA, most people choose to begin by generating a certificate signing request (CSR) through cPanel. When doing so, you're offered the option of generating a new private key or selecting an existing one, like so:
With an ACME CA (like Let's Encrypt), usually the ACME client generates the private key and CSR for you. You may never even see the CSR. For example, CertSage internally generates the CSR then throws it away once you have your certificate. Uploading a private key works like so:
If the private key is in the list in cPanel when you install your certificate, cPanel automatically pairs them. If not, you get an error. If you delete a private key, cPanel automatically deletes any CSR and/or certificate paired with that private key.
Note that you must install your certificate after uploading it. Simply uploading it doesn't automatically install it!
Please also note it isn't mandatory to let the ACME client generate the private key! It's in principle also possible to use the CSR generated by e.g. cPanel to be used in an ACME client. However, probably not every ACME client offers this option though.