I think your dealer's wrong about "battery saver mode", but the Gryphon does pull current when the truck is turned off.
I had somewhat the same experience when I got my original Edge/Gryphon back in 2007. After about 2 1/2 - 3 weeks, the battery would be dead. I finally did some checking on current draws when the truck was parked.
Once the truck has "sat" for about an hour, maybe less, the battery saver feature kicks in. But, there is still a 0.05 Amp draw (50 miliamps) even when this has taken place. The current is powering the Keep Alive Memory (KAM) in the radio, the clock, the wireless door lock receiver, the position of the driver's seat (in my truck), the A/C temperature and so on. There is also some KAM current going to the PCM that holds the long term fuel trims, and other "learned" portions of the "adaptive learning strategy" it employs.
By itself, that's not too bad of a current draw. It amounts to 1.2 Amps per day, and, if a battery is considered discharged after 55 Amps are gone, it can sit for 45-46 days before the battery is dead.
However, there's a much higher draw for a short period (until the battery saver kicks in) after shutdown and this high draw resumes as soon as you open a door.
Now the bad news: The Gryphon (older style) also pulls about 0.05 Amps of KAM so that it doesn't have to totally reboot when you restart the truck. I measured this draw and Bill confirmed that this was the design spec. So, with a Gryphon plugged in, you can only expect to leave the truck sitting about 21-22 days. (Much less if you open the door now and then).
My solution was to always connect a "Battery Tender" or "Battery Minder" when I pull into the garage, since my truck is used infrequently (for towing or trips). Camping World sells the Battery Tender and you can find the Battery Minder Plus online. They're both in the $40-$50 price range. They are much more than a "trickle charger" in that they won't "boil" your battery. And, the Battery Minder desulfinates the battery as well.
You can also disconnect the Gryphon when the truck is parked, but it's a pain to have to reconnect it after turning the ignition on when you want to use it.
I don't know if the CS/CTS units pull any current when the engine is off, so this may not apply to them.
Anyway, as Bill told me - the Gryphon was not designed with "Garage Queens" in mind!
Hope this helps.
- Jack