Solution
DHCP issues are rare, and the best way to resolve them is to simply format the RT target: Restore NI Linux Real-Time Target to Factory Default Configuration - NI.
You might restore the settings through NI MAX as well. This will require to connect to the controller directly, since it cannot be accessed through the network. Restoring the Linux RT target will reset the network settings and likely set it to DHCP as a default.
You might also try to directly connect to the PXIe controller through Ethernet cable. If this is possible, you should be able to set the IP address configuration to DHCP or Link Local in NI MAX. This method isn't foolproof, since the adapter might assign the IP address to Link Local while you're switching back to the local network.
If the device still cannot be pinged after being formatted, it's worth checking Wireshark packets. If the Ethernet packets are transmitted, chances are the issue is related to IT restrictions and should be resolved by the department.