Hi javajunkie,
If you graft a cat to a dog, the cat will make cats - if a branch from the dog is left to grow, it will make dog - They won't mix .
As for dropping figs, this can be climate related. So if your weather is too cold, most of the cultivars will drop the figs.
I know of this, because a friend has my same cultivars in a little more cold area and 2 cultivars drop everything, and the ufti some years she will produce maincrop only - with cold spring brebas get dropped - and some years brebas only.
One of the droppers may well be brunswick - but I still have to confirm .
But if your tree is dropping because she is too softy, then grafting a better strain can work - and that strain could produce figs.
Personally, I would get rid of the bad cultivar - but it depends of the size of the tree and if for you, it is worth keeping - for me it is not, but you are the judge.