One of the pros may recognize that handle and suggest a new stem for it. Usually, if you take it out and to a good plumbing supply shop, they can find a replacement for you. That means the water needs to be off while you're away, though.
But, it would be safer to replace the valve with a modern one that had the (today) required anti-scald technology. Those tile are soft, so easily cut to gain access and to then install a new valve. Most companies may a remodel plate that would then cover the larger hole in various shapes of chrome (and other metal finishes, if you wanted to change the rest of the items).