how should you charge a client for a new service in the middle of the month if they are billed on the first day of the month?

The best way to make sure customers only pay for the days they use a service is to add prorated billing to the agreement.

If you would like to have prorated billing (the invoice for a service is calculated according to the cost per day or the proportion of the monthly service used) you can do that when setting up the service, under the Billing & Payment section, as seen below: