My solution for this setup is having ingress controllers on all three nodes, and then specifying all three IPs in all DNS records. That way the end user will "load balance" based on the DNS randomization.
Of course, if a node goes down, a third of the traffic will be lost, but with low TTLs and some planning, you can minimoze the impact of this.
It's an interesting approach.
I did it a bit differently. I set up three Proxmox nodes on three hetzner servers. Then I deployed virtual routers. I then set up HAProxy and k3s nodes as LXC containers.
What's nice about the whole setup is that a proxmox node can go down and it all still works. I will now set up keepalived as mentioned in the other reply so the HAProxies will also be fully HA. Proxmox also works well with zfs and backups.
I set up the proxmox nodes manually and did the rest with terraform + ansible. One `terraform destroy` cleans up everything nicely.
I wonder how the performance difference is between bare metal and k8s node in LXC.
Of course, if a node goes down, a third of the traffic will be lost, but with low TTLs and some planning, you can minimoze the impact of this.