Include the new Maverick JavaApplicationStub in the installer. Have installer do a system check condition. If OSX 10.9 or later, install the Phex app package and replace the JavaAppStub with the new Maverick JavaApplicationStub within the app package on disk.

The problem is definitely the JavaApplicationStub file not being compatible with Mavericks.

Would probably also be good to include upgraded settings such as I have in the connection installers. ie: port blocks and host file size.
My connection installers insert the text into the settings file if the properties file exists. Otherwise a fresh properties file is installed.

My Linux connection installer script includes the upgraded settings which should be readable on Windows or Linux. The following link also includes the Mavericks JavaappStub https://www.mediafire.com/folder/xbqablpc3sttu/Phex
Just discovered one of my hosting sites has recently closed down (Hotfile.) In several hours later when I get time I will shift some items to Mediafire instead. (Some items were only hosted at Hotfile, such as the LW Clone connection installer.)

* Edit: After having woken up a little more. A better approach would be for installer check system, if OSX 10.7 or later, install phex app package, then replace the JavaApplicationStub with a copy-replace of the one on that person's system. Does that sound messy to you?
Alternatively just install the application package without the stub file and duplicate the specific system's stub file & copy to the app package MacOS folder. In that way there's no replacing & no system checks.
Destination of the JavaApplicationStub would be the user's chosen location for the phex app package. Phex/Phex.app/Contents/MacOS or Phex.app/Contents/MacOS

The snapshots folder is a different kettle of fish. You might need to include two separate phex's for OSX (one with Mavericks stub file.)