PvPvE. (for some that might not get it for some reason, Player versus Player versus Environment)
Make it a simple deathmatch, or add vehicles, I don't care. Look at it as Big Team Battle from Halo if you were to compare it. Make the map each individual area of the planets or moon, sure set some boundaries. However, throw in some events where a random group of Fallen, Vex, Cabal or Hive come into play and it turns the pvp training into a kill the enemies match, each enemy killed, based on their threat level gives said team a point(s) on who got the final killshot or damage percentage. Drop down a devilwalker sure. Or throw in a randomly generated boss and turn it into a short term horde then resume pvp functionality (pvp combat would turn off while the enemies invade). It would combine a large set of matchtypes while being somewhat unique to the genre, at least a refresher.
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I'm surprised noone has suggested a multiteam mode. Since the max player count is 12 it could be a mix of 4v4v4 or 3v3v3v3. Either 3 or 4 teams:) skirmish & salvage are fun but they are too limited to one fireteam vs another. Another gamemode idea could be racing with the sparrows checkpoint to checkpoint... the point is to add variety to the players.