What an unfortunate predicament you are in (having an ultimatum or else...)
You are right to be concerned about your 6 yol granddaughter. Children at this age are just starting to see the outcome of inappropriate behaviors. If mom does not want the child to engage in certain behaviors, threatening is not the best way to role model how to act.
You are also in a tight corner with your son not supporting you and as you mentioned quoting scriptures. If the family is quite religious, (I do not know where you stand in this, but you can remind them that
"Love is patient and kind; love is not jealous or boastful; it is not arrogant or rude. Love does not insist on its own way; it is not irritable or resentful; it does not rejoice at wrong, but rejoices in the right. Love bears all things, believes all things, hopes all things, endures all things. Love never ends.
(Bible, part 1 Cor 13)" And that this applies to how one interacts with others including their own mother and daughter.
Here I attached a link to the moral stages of development and Nan, based on her age will fall into the fist one.