The client certificate is sent by the browser only in case of HTTPS. Therefore, you first need to install the server certificate on your web server in order for HTTPS to work.
Then I do not think that you can force the use of SSL and a client certificate on the same page, but you can create a subdirectory of your site where you enable SSL and require a client certificate. Then you put only your special registration page in this subdirectory and link to it using your registration button.
The rest of your site may be anonymous HTTP.
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