Hi, I am new to Clarity and like to answer this question although it is 8 months old. I had a similar challenge in my online marketing agency which I solved with technology.

I used Asana to manage each task on a customer account, even my meetings, phone calls was added as tasks. My team members had to account for all their time using assigned Asana tasks.
For timekeeping I used Harvest Time which integrates perfectly with Asana. Team members was able to manage all their time from within Asana with the Harvest timer button.

In Harvest Time I could track all time spend on each customer and if it reached it pre-set limit I would contact the customer to negotiate more budget or to move all new tasks over to next month.

This changed my business profitability big time, especially with Harvest Time project view.