I agree that clergy should be willing to pray at ceremonies for elected officials if asked.

Without having the details, it is impossible to know for sure, but what seems problematic in this case is that Cardinal Dolan may be playing a supporting role to Paula White. Apparently, she is giving the invocation and Dolan and one of the others will do readings. We don't know who choses the readings, let alone what they are. 

In this situation, there is considerable potential for Cardinal Dolan to be embarrassed by his association with the self-declared pastor of a church she and her husband founded. None of the Christian clergy (except Dolan) are associated with large or mainline denominations.