You are right, running_sum() does work perfectly with a normal filter. I realized my attached workbook was actually an oversimplification of the problem I am experiencing. In the case of using a parameter as a filter, running_sum() ignores the parameter.
I did solve my problem with your help, by creating two calculated fields:
Check if date matches parameter filter
IF DATE(LOOKUP(ATTR([Date]),0)) = [Date Parameter]
Running Sum of parameter measures only
Glad to help. If you don't have further question, can you mark my answer either as correct or helpful to close the thread? ta