One among the few shops in London that offer or offers designer clothes.
It really comes down to whether you are referring to "one... which offers" or "the few... which offer". From your example, the actual focus of the sentence is somewhat ambiguous. More context would give better clues, for example:
"One among the few shops in London that offers designer clothes is located on Main Street" if you are referring to a specific shop in this category.
"One among the few shops in London that offer designer clothes may hold the item you're after" if you are referring to all such shops collectively.