Solve Chicken and egg
You'll have to solve the chicken and egg problem when saving a new Customer. You have to save to get an id, but you have to know the id to know where to save.
You can solve that by saving all Customers in DatabaseA first and then check the id and save it in the target db too. See Django multidb: write to multiple databases. If you do it consequently, you won't run into these problems. But make sure to pay attention to deleting Customers.
Then route using **hints
The routing problem that's left is pretty straight forward if an instance
is in the hints. Either it is a Customer and you'll return 'DatabaseA' or it has a customer and you'll decide on its customer_id or customer.id.
Try and remember, there is no spoon.
When there is no instance in the hints, but it is a model from your app, raise an error, so you can change the code that created the Queryset. You should always provide hints, when they aren't added automatically.
What will really bake your cookie
If for most queries you have a know Customer, this is ok. But think about queries like TableA.objects.filter(customer__name__startswith='foo')