Either you need to install both certificate and private key at the load balancer or none of these.
You need to install both if the load balancer should also to terminate the SSL connection. Such termination is for example done for performance reasons in which case the SSL connection is terminated at the load balancer and forwarded as plain to the final server. Given that you've installed certificate and key at the final server (which would not be needed if it gets only the plain traffic) this is probably not the case in your situation. But it might also be needed for the load balancer to make decision based on the application content, for example to make sure that all traffic with the same cookie always ends up at the same server. In this case the server needs to terminate the SSL traffic too in order to get access to the content and then might talk to the final server either in plain or re-encrypt the traffic using a new SSL connection.
If the load balancer is instead just balancing the traffic based on layer 3/4 information (i.e. source IP+port, destination IP+port) or if it balances based on the
server_name extension (SNI) in the ClientHello of the TLS handshake then it does not need to decrypt the traffic which means that it needs neither the certificate nor the private key.