I don't believe this is possible if the Caller ID is making it into the system from the telco, and with a PRI (assumption since you said it was a DID number) you really can't have them turn it off, it is part of the ISDN signalling format.
So you have two options as I see it:
1) Loop a few single line ports back to LS trunk ports, put the stations and trunks in the appropriate groups, point the DID to the Hunt Group, then ring the trunk group to the original DID's destination. You can turn off Caller ID on the LS trunks easily and they will see the Trunk Group name instead.
2) Get a new block of analog CO trunks, connect to the system, and either move the DID to a copper line or call forward it via a phantom to the new CO trunks hunt pilot number.