Can't help you as I see no point to adding a laundry detergent 'filler' (sodium sulphate) to bar soap. The purpose of it in commercial laundry detergents is stated as a filler, therefore it's only purpose is to increase the volume of the laundry powder so the consumer things they are getting more product. In my opinion that is a waste of resources and ethically questionable.
I have not read anywhere else where anyone has used washing soda (sodium carbonate) in bar soap, but I do use it in my powdered laundry soap as well as my laundry butter. It's purpose is two-fold: stain removal and softening of hard water. It is also used in liquid laundry soap. But as I said, I have no idea how it would affect solid soap.
Maybe you can do some test batches with varying amounts and see what results you get.
Here is an experiment done by a formerly active member about 8 years ago using sodium carbonate vs potassium carbonate (as a point of interest, but it probably won't help you determine how much to use):
http://www.soapmakingforum.com/showthread.php?t=13704
And here is a link about laundry detergent formulation that might interest you:
http://surfactantschemistry.blogspot.com/2013/07/laundry-detergents-formulation.html