Hi Oliver,
I gave it thought and realized that the project is is already having the % Completion Calculation method should be selected as "Manual", else you would have not been able to edit those fields under the Task Properties Page.
However, the status is not going to change to "Completed" automatically, because there must be some ETC for the resource or the task. Please set the ETC to 0, Change the percentage to something less that 100% (for example 98%), and then change it back to 100% and hit SAVE. The Status should now change to "Completed", by the system itself.
Hope that answers. PPM will not mark a task as Completed if it sees ETC left for the same, which means, there is some work left for the work to be completed.
Cheers!