That won't work. Let's Encrypt issues Domain Validation (DV) certificates, meaning only the domain ownership is verified. Other fields will be discarded before Let's Encrypt signs the certificate because Let's Encrypt has no way of knowing if that data is accurate. What you're asking for would be a Organization Validation (OV) or Extended Validation (EV) certificate, which involves a manual validation process and is not available through Let's Encrypt.
When you manually provide a CSR file, you would have already had to create a private key in order to do that, so the file should already exist somewhere. You probably used something like openssl genrsa ... to generate that file.
The CSR does not contain your private key, so the client has no way of storing the key in a file of your choice. That would be up to you.
For the previously mentioned reasons, that would not be possible.