Use below tool on Linux based systems to check ICMP (ping) connectivity:
traceroute -n <IP>
For Windows you have similar tool:
tracert -d <IP>
Run below one on Pi-hole and on the clients (Linux/Windows/MacOS) to check DNS port 53 UDP connectivity using the OS configured DNS server(s):
nslookup pi.hole
Run below one on Pi-hole and clients to address a specified DNS server:
nslookup pi.hole <DNS_SERVER_IP>
Pi-hole only does DNS filtering and as you might have noticed, above connectivity tests are all IP based (not domain name based).
So if have issues with any of above ones, you have to check your VLAN setup and maybe route 53 UDP & TCP between the VLAN's.
Or you could assign multiple VLAN's to the Pi-hole port on the switch and let Pi-hole have a foot in all the net segments (multi homed).
Checkout below posting how to get as many IP's configured (IP aliasing) as you like on one physical (or virtual) interface: