Can you use a name brand product for a base in your barbecue sauce to sell to the public without any fines or breaking the laws?

No, in making your own sauce for public sale, you cannot use a product owned by another without permission. The owner of that product not only has rights in the sauce itself but in the branding associated with it. As such, using it may violate multiple rights that the owner has in the product. With respect to the FDA, their role in marketing of your proposed product is primarily a regulatory one, whereby they regulate the actual manufacture of the product in order to ensure a safe product for the consuming public. In other words, the FDA does not handle the legal aspects associated with trade secrets, trademarks, etc.

Not sure I understand your question - but I would caution you that as far as I know, all consumable or edible foods sold to the public must be marked as to their contents. In this way anyone who has a severe egg or nut allergy etc would be advised. However this is not my area of law. You need a lawyer that works in FDA regulation.

I believe the FDA requires labeling to disclose the specific ingredients (of the types where disclosure is required). You need to research it yourself or hire a consultant with experience or knowledge in FDA certification.

Recipes are often considered trade secrets. Altering a recipe without prior permission of the owner, especially of a name brand product, is impermissible. Labeling such a sauce with your own name, without the owner's prior permission, would also violate trademark law. And the cost of using the name brand product as a base for your product is going to be much, much more expensive than inventing your own sauce base.