You cannot download the certificate from the dashboard. You’ll have to do so from the API using the /client/v4/zones/<zone_id>/client_certificates endpoint (or pick it up from the network tab in developer tools).
However, the certificate itself is not very useful without the private key. When you create a client certificate in the dashboard, your browser generates the public/private key-pair used for the certificate. This is the only time you can access the private key. The private key is discarded when you navigate away without ever being sent to Cloudflare.
You cannot create a CSR without a private key. That means you had one on the device used to generate the CSR. Using a certificate management tool can make it easier to track the relevant components. I liked XCA from the first time I used it, so I haven’t spent any time looking for alternatives, but it is not the only option.