I think the root cause is that unlike the bridges and tunnels into Manhattan where the tolling is done primarily via transponder with plate reads as a fallback, the CRZ fee is captured by plate reads only. As posted above, the only way to get the $3 crossing credit is a matched posting to a personal EZ Pass account with a Lincoln or Holland Tunnel toll and a Congestion Zone entry. The toll posted to Avis' vendor isn't matching with the plate capture (which appears to be done by PlatePass).

The solution is to use a personal transponder linked to an account that allows for temporary registration of a rental car (note that NY just enabled this feature about a month ago).