For that to work, your nginx has to correctly handle the forward to http://<2nd-pihole-host>/admin.
Note that you have to specify /admin for access via IP address.
It would work, but you'd end up on the UI of the Pi-hole that the requesting client would have used for DNS, as that would have translated pi.hole to its IP.
Your debug log shows that you have enabled Conditional Forwarding to allow local resolution through your router's DNS resolver, but you haven't configured a local domain name to go along with it.
Without that detail, Pi-hole would forward a client's requests for <hostname>.<search.domain> to its public upstreams, which naturally would not know anything about your local private network.
A failure to resolve may also suggest routing issues, i.e. some VLANs may not be allowed inter-VLAN communications to Pi-hole.
To analyse further how DNS is involved, use dig or nslookup from a client you suspect to have DNS issues.