Robotic button pusher
I'm in New Zealand - just across the pond. How many of these can I sell? :-)

GEP manual here - no indication of remote controllability.

ebay = GEP for sale - $US895 

It seems unbelievable that the manufacturers would not have made these units "remoteable" but a quick web search does not reveal any information that suggests that they can be remotely controlled. Making it so should "just be a matter of engineering" [tm].

Two main methods suggest themself.

You could make a completely external button pusher using 7 solenoids or servos mounted on a frame which was able o be located rigidly relative to the GEP. This is clunky and Heath Robinson but would work and leaves the unit unopened for resale and warrant purposes. 

The "proper" solution would be to make electrical contact with the joystick/button connections and operate parallel electrical switches remotely. This is liable to be very easy conceptually and in practice anywhere between not too hard and stunningly difficult, The latter is extremely unlikely, but designers sometimes do very very obscure or unusual things. So it's best mentioned, just in case.

Actual remoting in either case can be by wore or wireless in a range of flavors (Bluetooth, WiFi, ISM band short range, ...) 

If you need the LCD display remoting , again two main methods suggest themselves. The Mickey Mouse Heath Robinson clunky method is to use a small video camera (web cam level) to view the display and drive a remote monitor. This is liable to be very satisfactory if done even half well and MAY be much much easier than accessing the LCD signals electrically. LCD signal access may be easy and may be diabolical. Worst case they may use some form of COB (chip on board) on LCD glass driver with no leads accessible and for an extra bonus could throw in a processor as an integrated part of the LCD. This is an unlikely approach but COB is quite conceivable. If they do use integrated driver you may be able to access the LCD drive data stream which could be very easy to remote and to decode. Or not. The webcam approach does have advantages and is a nice fallback position if you do decide to try direct access and fail.