Might not be able to do this, as the router and or firewall might have the ability to block DNS traffic. They might be able to setup a rule to block all traffic on port 53 except for the DNS servers they want you to use.
Keep in mind that nslookup is a Unix utility and might not act like most MacOS apps which use the special MacOS DNS Resolver service. I wouldn't make any assumptions based on how nslookup behaves. How are you trying to change the DNS server manually?
Might not be able to do this, as the router and or firewall might have the ability to block DNS traffic. They might be able to setup a rule to block all traffic on port 53 except for the DNS servers they want you to use.