If you do, it is 100% justified.
However, I understand your position when it comes to completing scheduled work, regardless of employee indiscretion.
Please do not take this as a boast, because it’s not. But I’ve managed a string of restaurants with hundreds of employees before turning to window cleaning. Believe me when I tell you: I’ve seen it all when it comes to employee indiscretion.
I’m going against the grain here with my response, but, so be it. There are plenty of successful businessmen on this forum and you may take their advice with no fear of reprisal.
What I will advise you is this: SOMETIMES the employee you catch red-handed can become your best. I had an experience similar to yours with an employee who ripped off a customer. I gave him a chance to confess, and he did. Recompense was made at his expense, and he is still employed at the establishment to the best of my knowledge. He was one of the hardest workers. And one of the most reliable. Following said incident, he was THE hardest worker, and THE most reliable. He cared about his job, but he got stupid.
You can toss your guy like a sack of trash, but just be careful not to shoot yourself in the foot. It sounds like you are up against the wall when it comes to completing your scheduled work. If there is ANY hope of rehab, I suggest you at least make the attempt. Give him a chance to come clean (if he is as yet unaware of your conversation with the customer) and mandate that he personally refund the cash to your client, with a verbal apology in your presence. If he is unwilling to perform this act, then can him.
If you believe it will be a simpler matter to replace rather than rehabilitate this individual, then can him.
You may opt to wait until you are through the heavy season, restrict this employee’s autonomy, train a new staff member, and then let the perpetrator go his own way.
Ultimately, it is your responsibility as proprietor to do what YOU believe is best for the business. And no one can fault you for that, regardless of your decision.