No, you cannot change a studio's URL. You can rename a Studio, but this does not affect the URL. A constant URL allows any invite links you have already sent to remain valid since they use the URL that corresponds with the Studio's original name.
Riverside for Business users can create a new Studio with a different name, which will be used as its URL. For example, if you call your new Studio: Possible Barefoot Moose?, its invite URL will be something like: https://riverside.fm/studio/possible-barefoot-moose?t=0123456789abcdefghij.