Is the reason that my 9.5 battery only operates to 3kw 100% because it has been fitted with a 3kw inverter?
No, the 3kW on the inverter refers to the maximum charge and discharge rate of your inverter, not your battery capacity.
If you are not getting full capacity and its a new install then I would get your installer to come and look at it. It sounds like it needs calibrating which your installer should have done on installation