Should I just keep overwriting the existing skeletal mesh asset when I export and not worry about versions?
Yes. If you want to keep a history of versions you can “reverse” the version numbers when saving from Maya (easiest to do through script because file renaming is involved) so that the latest always has the suffix removed (character). The next oldest would start as version 1 (character_001). This way you can always right click > Re-Import from the content browser with the latest update.
If possible, its best to let a source control system (like Perforce) handle your version numbers under the hood. There are many additional benefits to using source control in your projects, here are the docs on how to set it up…