If you've never visited GaP before, Edelweiss would be my last choice, regardless of eligibility. Unless you're among the lower enlisted ranks, it isn't that cheap and it sits outside of town, surrounded by a high wall. Looks kind of like a prison from the outside.
Anyway, if budget is no concern, the nicest hotel in town (the Partenkirchen side, at least) is probably the Atlas Grand Hotel. If you stay here, try to request the Ludwig or Sissi suites. For a cheaper option just down the street, Gasthof Fraundorfer has comfortable rooms and a fun restaurant downstairs.
On the Garmisch side, I've stayed at Hotel Wittelsbacher Hof. Nothing wrong with this hotel, although the decor could use an update. But the location is very convenient to the Zugspitzbahn, the Kurpark (nightly concerts in the summer!), and the pedestrian shopping street.