A British woman’s claims that she contracted Legionnaires’ disease during a visit to a hotel operated by Elvis Presley Enterprises Inc. will proceed, a federal court in Tennessee ruled.

Santa Barbara Control Systems, a water treatment provider for the pool and hot tub at the Guest House at Graceland LLC, argued that there was no proof that its products played a role in Jennifer Walker’s illness. But the U.S. District Court for the Western District of Tennessee rejected the company’s...