I suspect the 2960X might work as the 3750 series did when it did allow different IOS images to run on different stack members, with a correctly functioning stack. What would NOT work was if the different IOSs had different, major version numbers, stack code (not to be confused with IOS version numbers). I recall if there was a difference in the minor version numbers, again for the stack code, device would throw a log message, during boot up.
A good rule of thumb was to insure all members were running the same IOS release (although, again on the 3750 series, they also did not need to have the same feature sets).
I believe later series Cisco stackable switches are more demanding requiring all stack members run identical code with identical features (licensing) too.
There's also, I believe, support to automatically push a stack's running code to a booting stack member without that image. (I believe I've read the later series [or later IOSs?] handle dynamically loading a booting stack member's running software much better.)
Since you've already have a running stack member, I would presume there's still a way to update an IOS (and bootvar) on a specific stack member, and if necessary reload just that stack member.
Or, confirm whether your stack supports the dynamic push of an IOS to a booting stack member, and if it does, determine why it's not happening.
Lastly, if you "install" an IOS onto the stack, I believe it's supposed to make all necessary changes across all the stack members.
