As there are no issues other than fussiness why bring up possible issues to say the guy was wrong?
Forgive me if I am wrong, but I read it as the owner was a woman?
She didn't know that though did she? The owner just presumed and was IMO quite confrontational, the tone conveyed in the op reads as rather hostile and abrupt to me and OP clearly states that the conversation was loud enough for other customers to hear.
Quite frankly as an independent burger bar which are dime a dozen and most likely rely on repeat customers I would have thought that even when laying down rules you do so with a measured approach and in a way that doesn't put the customer on the wrong foot yet clear enough to be understood and most importantly of all polite and discreet.
OP did buy food from them so its not like the burger bar didn't make any money from them, they could have just as easily waited until the end of the meal, its less confrontational that way, raising the issue whilst they were eating just means that the whole meal is awkward, I doubt any other customers were drastically affected by a 9 year old girl eating a sandwich whilst her family ate food from the bar.
You are right, it is her burger bar so she can do as she wishes, but They've just lost a customer, I am surprised at her lack of business sense, I would rather have a repeat customer who spends XX amount and put up with her child eating a simple sandwich than alienate the whole family, lose them as potential revenue and risk them bad mouthing my business.
Its an independent restaurant, from my experience they have a little more leeway with what they class as acceptable than a chain.