I guess SSL::c3d might be a start.
Otherwise, you might need to change both the client and server-side SSL certificates to ones configured for using C3D, and then renegotiate SSL on both the client and server side.
Just by way of explanation, C3D allows the details in a client auth certificate to be passed to the server-side SSL negotiation, by creating (forging) a new and valid server-side client authentication certificate based on the server certificate for the server-side TLS negotiation. This is only necessary when the pool member SSL stack needs to see those specific client auth certificate details (for authentication/auditing purposes). If you can get away with a single shared server-side client auth certificate, it's usually an easier option.
I haven't played much with C3D, so it's a bit of a speculative approach, but it's where I would start.