Personal opinion: The City of London officer who fired the shots had a split-second to make an assessment. Sees a vest, guy moves, bang. Neutralise the threat. Notwithstanding it being hoax or not, I believe the rules of engagement for police, especially anti-terrorist, is that it is better to shoot dead than to risk more casualties. A bomb going off on the bridge would have meant more lives lost, more damage done, let alone maybe causing structural damage to the bridge itself.
They would have loved to keep him alive. It will be lauded by extremists as martyrdom-by-cop and it's entirely probable that he wore the vest to ensure he would be taken out and having "died for the cause".
One other thing comes out of this. There are still ordinary people willing to run towards and fight danger. I hope all involved, police and public, get the recognition they deserve at preventing larger loss of life.