Hi David, How are you doing? I have missed you guys, too.
Hi, Grammarcrazed: I'm doing well. Thank you for asking. Adapting to this new Web platform has been a bumpy ride, but things are looking up, overall.
So, would you label (b) ungrammatical?
I'd label it incorrect. Labeling it ungrammatical would be going a bit too far, I think, since the descriptive genitive does exist ("He came back from the store with more lady's gloves / men's shoes / children's clothing" -- examples taken from George Curme's 1931 grammar) and there is no reason, in principle, that burger joints shouldn't follow that pattern. However, the fact of the matter is that they don't. The use of "Burger King's burgers" in your example (b) ("He came back with more Burger King's burgers") is a non-conventional descriptive genitive, and is so unidiomatic that it may be deemed incorrect.
I suspect that the culprit here is business names like Macy's and McDonald's, which are themselves possessive. It is correct to say "He came back with more McDonald's burgers" or "He came back with more Macy's shirts", in which sentences the name of the business, which is always possessive, is functioning as an attributive noun, not as a descriptive genitive. (The possessive of McDonald's, according to Bryan Garner, is technically McDonald's's.) As you know, there is nothing inherently wrong with the phrase Burger King's burgers, which may be correctly used as a normal (non-descriptive) possessive: "He likes Burger King's burgers." But the attributive construction also works: "He likes Burger King burgers."