So, you know your handset supports CWCID (Call Waiting Caller ID).
That being the case, here are a few settings you can look at:
CallWaitingCallerIDEnable, set to enabled
CallerIDDetectMethod, set to default, assuming you are in the US or Canada
CallerIDMethod, set to default, assuming you are in the US or Canada
CallerIDTrigger, set to default, After First Ring, assuming you are in the US or Canada.
You'll find a reference to these
here, search for each one by the above names.
The last setting in the list only applies to the first (on-hook) caller ID, not the CWCID. But check in any case.