That is a tough call because there was 437 actually recorded people that are known to have worked on the bridge and someone responsible would probably have crossed in order for the bridge to actually open the bridge. Surely there were more who crossed known or unknown. There would have had to have been actual vehicles crossing as well before the bridge was opened as well.
In order to confirm this you would need speak to the historian for the bridge or the record keeper(assuming the bridge actually awarded Clarence). Beyond that starting at your local library and searching their archives around the time the bridge opened will give you information regarding this.
What I would do as an actual starting point is either querying the actual historians of the bridge itself, those responsible for the upkeep and maintenance of records.
The general line for the Tower Bridge is +44 (0) 20 7403 3761. I would start here and request information or the contact information from the history keeper or archive maintenance people in order to confirm or deny.
The problem is the formality of the actual event if it did occur. How many actual records were kept, who kept them and where as well who or what actually awarded him the silver whip. Chances are the electronic transfer of this information is going to be severely
limited as it probably isn't wasn't considered essential to the history of the bridge, its build and maintenance, so chances are if you are going to find it, its going be searching and finding it in archives in person.
One thing I will tell you is to be very persistent in your search. Old records(the physical type) are notorious tough to dig up and secure.