If that's the case, then it just got pretty complicated. We have a mixture of SIP and PRI and the way we have it right now is that particular users DID is on SIP but the outgoing route is with the PRI with the outgoing ID from the SIP. Who do I chase on this case? Would this fall on the PRI since that's where we the outgoing call route is?
That isn't really how it works... When a call goes out, it would go over your PRI or your SIP connection. If you are sending Caller ID of the SIP DID out your PRI, then your PRI carrier would need to first allow a call from a number it does not have provisioned on your account (many do not allow this and Stir/Shaken would be invalid, which is also a gray legality)... Some will allow the call, but substitute the billing number of the PRI instead of the CID sent with the call (the billing number may or may not be the "main" number), meaning the number you sent as the Caller ID doesn't matter.
How Caller ID Name traditionally works, is when the terminating carrier delivers the call to the callee, it does a dip to a Caller ID database of the Caller ID ANI number it receives and if there is a match it sends that CNAME info to the callee. In this case your SIP provider "owns" the number and they would have to update the CNAME associated with that DID in the database. However, some SIP providers use a more "modern" approach and allow CNAME passthrough (meaning they send CNAME based on what you send with the call). If they just do passthrough and don't update the Caller ID database, then if you send the number out the PRI, no one really "controls" the name associated with it and most PRI providers do not support receiving CNAME from the end user, so either no CNAME is displayed or the name associated with the primary billing TN is displayed.
Clear as mud, right?