Well done Keybright. I think you did the right thing.
Food for thought though: The probationary period can sometimes be a double edged sword.
As has been mentioned, you do not have to give a reason to 'let an employee go' within the first 12 months of continous employment. However, the introduction of a probationary period into an employees T&C's forms a seperate part of an employment contract. The employee may well be within their rights to insist on a reasonable explanation as to why you are letting them go. "Sorry, but things have not worked out" may simply not be enough.
Although this would not constitute grounds for suing at an employment tribunal, a disgruntled and determined ex-employee could certainly use it to stir things up for you.
When ACAS tell you that you should follow the verbal and written warning procedure first, they are simply trying to advise you in a way that is beneficial to the employee, as well as you the employer. They are not saying that you should do this as a matter of law. Also, the warnings procedure works well for employees who commit minor infringements, but there is clear legislation on the side of the employer relating to instant dismissal for instances of serious misconduct.
The only issue then is, what constitutes serious misconduct. If the attitude or behaviour of an employee can be shown to have lost a business three clients, then that employee has put the wellbeing of the business in jeopardy. This easily constitutes serious misconduct.
Taking into account your employee was within the probationary period and the fact that you do not normally need a reason to 'let an employee go' in the first 12 months, I would still consider using serious misconduct as the reason in your situation. You might find that the employee is less likely to kick up a fuss because a valid and serious reason was given in any event.
I sincerely hope this is the last you hear of this person and I wish you all the best for the future.