Is Biden trying to get a conservative journalist's press privilege revoked?

From The Hill newspaper:

Joe Biden’s office has complained to the Senate press gallery about a confrontation the vice president had with a conservative journalist last week on Capitol Hill.

Biden aides asked whether Senate rules were broken in the wake of the contentious exchange between the vice president and the reporter.

Jason Mattera, who works for Human Events, a conservative magazine, used a pretext to catch Biden off guard in a Senate hallway and grill him on claims the vice president has made about jobs legislation.
Biden’s office has also contacted the standing committee of correspondents, which oversees the gallery, regarding whether Mattera broke the rules by ambushing him.

Heather Rothman, the chairwoman of the gallery’s standing committee, said the matter is under review. . . . .

The ruckus seems to be over the fact that the reporter shook Biden's hand, somehow deceiving Biden into believing that the reporter was just a normal citizen asking questions. After watching the video, there are a couple of points. There were lots of press hanging around there, including movie cameras. Whether it was a reporter or a private citizen asking questions, Biden would have known that he was on the record.

Would the media be all over this if it involved a "main stream" reporter?