You could try a ciabatta / focaccia type roll? Not entirely what I would call a burger bap, but I rather like them.

And if we are talking baps (perfectly normal phrase for a filled roll for lunch!) what about barms, barmcakes, buns, stotties, yorkshire teacakes (not currant teacakes), cobs, batches, oven bottoms, etc. Many a night in halls at Uni spent arguing over proper terms!