You'll definitely need to use a different method for getting the totals. But you may not need to do as much data copying as you are currently doing and the change that I am about to suggest can simplify how to get your totals.
Instead of copying the data, link to it. Each of the fields that form your columns of data, one column for each employee can be calculation fields that compute a total from the table where this data is recorded. The calculation fields can each use a different set of match fields to match to the correct set of records such that Sum ( RelatedTable::Field ) will compute the needed total, or (if you are using FileMaker 12 or newer) you can use ExecuteSQL to access that info. With either method, you'd have two pairs of fields for each such column of data. In one field you select the EmployeeID of the employee whose data will appear in that column. The other field is the calculation field that computes the needed total. The table where you set this up would then need to have one record for each client in order to set up your "One row for each client" format. In any given record, each of your employee fields uses the employeeID specified for that column and the clientID specified for that record to access the needed group of related records used to produce your total.
Your summary totals can now be computed from a set of summary fields, One Summary field for each column of Employee totals.