Yes it sounds to me like Filemaker will do exactly what you need.
Firstly you will be best to sit down and have a think on exactly what you want and need to hold data on before starting off.
As a starting point it sounds like you may need;
1) a client table holding the basic client information
2) an employer table (related to the client table)
3) a funding source table (related to the client table?)
4) a client timesheet record for the activities (related to the client table)
You can then draw out some portals to show the related records, allow access to the related records (if needed) and setup some reports.
Hope this helps - even a little!
P.S. The trial allows you to do everything the full (pro not advanced) version does!