Even though resolv.conf has
"search cattle-monitoring-system.svc.cluster.local svc.cluster.local cluster.local"
it just doesn't search any of them and just fails.
Other containers can perform the exact same nslookups, but the pushprox-k3s-server-client fails.
I also found a seemingly old thread about a similar Alpine bug, but as far as I understand from my containers, the bug persists in the 3.13.5 that I have.