Is there a clean way to have non-ringing phones press a feature and inspect Caller ID for the inbound calls that are ringing on other phones?

You could try the ispy key, can't what it does exactly though

make phone ring, turn off ringer and make ICM voice announce.

Not sure if it will work for a ringing trunk but check out the Inquire button, it allows you to view CID of a call and have the option of answering it or not. It may only work for parked or held calls though.
Nameless:
The ISPY button is an investigate button, similar to a barge in that it allows you to see who a station is talking to and view the number dialed or incoming CLI then you are given the option to barge in.

I tried INSPECT, it just says "Call No Longer Available" whether the call is still ringing or answered.

That would be the perfect feature for this customer if we could get it to work.