I think Gregory's answer is the best option: forget about the which/that dilemma and just add "...in London."
Maybe we can use a clearer version to show the difference:
"I got it at a vintage store which specializes in 80s fashion."
"I got it at a vintage store that specializes in 80s fashion."
These are both correct and natural, and mean exactly the same thing. As other commenters have correctly noted, "that" as part of a relative clause cannot follow a comma.
"I got it at the store's Charing Cross branch, which specialises in 80s fashion." Only this is correct. Notice that we can remove the whole relative clause, and we still know exactly which shop you mean.