Linked below is a post about using Bragg's in canning (not the recipe you mentioned, but plenty of discussion of usage).

Here is a link that might be useful: Using Vinegar with the Mother

I was going to post the same thread malna did and I think it answers your questions. It would seem that the taste 'might' be improved but the appearance would suffer. And yes you might as well strain the mother because the processing is going to kill it anyway.

As for subbing regular Heinz apple cider vinegar in Dilly Beans, yes I do so as I much prefer its flavor over white. We use cider vinegar rather than white in most all our pickling recipes.

Dave