I still think your better off getting the right airframe. Why modify a perfectly good 605 which will most certainly decrease it's value? Buy an early GLEX which can start the APU at 37,000 feet and operate up to 45,000 feet. Maybe a G-IV might be an option also, I'm not as familiar with Gulfstreams.
The reason the 605 can't run the APU at altitude is due to the inlet design. There is no door, your going to have to engineer a new intake and system to control it, and integrate all that into the APU FADEC. You'll see that aircraft that can run the APU at altitude have a APU intake door that can vary the opening angle. It's not a small project.
Even with that, I don't see the reason to need an APU. All your going to get from an APU at high altitudes is electrics, there are probably better routes to getting more electrical power than running the APU.
You can see many GLEXs modified for special missions, I can't think of any 605s. The 605 isn't that capable of an aircraft, it serves it's purpose fine but there isn't much extra in any of the systems for "special missions". Probably the reason you don't see any modified ones.