I have logged an enhancement request on our internal tracker. We will consider adding support for updating/refreshing WebTiledLayer in the future, now that there's a specific use-case for this extra functionality. Thank you for the suggestion!
It wouldn't be quite as simple as changing URL on new requests -- there's also cache invalidation, canceling in-flight requests, and redrawing involved -- but it's absolutely doable. This scenario just isn't implemented in Runtime yet.
Today, your best bet is to create different WebTiledLayers for different templates.