WAY : 1

If you are using Kubernetes cluster on GKE then you can use ingress indirectly it will create the Loadbalancer indirectly.

You can add SSL certificate store it inside secret and apply to ingress. For SSL you can also choose another approach to install cert-manager on GKE.

If you want to setup nginx-ingress with cert-manager you can follow this guide also :

https://www.digitalocean.com/community/tutorials/how-to-set-up-an-nginx-ingress-with-cert-manager-on-digitalocean-kubernetes

WAY : 2

Edit the helm chart locally add the liveness & readinesss probe to deployment and it will check wordpress health checkup over http only.

Update :

To add x-forwarded-proto in ingress you can use this annotation

nginx.ingress.kubernetes.io/server-snippet: |
      location /service {
        proxy_set_header X-Forwarded-Proto https;
      }

As the HTTPS load balancer terminates the client SSL/TLS session at the LB, you would need to configure HTTPS between the load balancer and your application (wordpress). Health checks use HTTP by default, to use HTTPS health checks with your backend services, the backend services would also require their own SSL/TLS certificate(See #4 of HTTP load balancing which HTTPS load balancing inherits). To make the backend certificates simpler to configure, you can use self-signed certificates, which do not interfere with any client <-> load balancer encryption as the client session is terminated at the LB.

You can of course use HTTP health checks (less configuring!) for your backend(s), this will not cause any client traffic encryption issues, as it only affects the health check and not the data being sent to your application.