Extend the schedule by adding Wednesday games, probably an extension to 10 months/year, while also condensing the playoffs - each home/away series should be a Wed/Sat format so that it can be completed as fast as possible.It would probably be more Wednesday games not a longer season.
They should not do away with a home/away format during the regular season for teams in the same conference. That would be annoying from an attendance standpoint depending on whether teams suck and/or have players worth watching.
The Championship has 24 teams, so 46 league matches plus a few playoffs for promotion. If they can play that many, there's no reason MLS can't do it too:
32 teams, 2 conferences of 16 = 3o games against conference opponents + 16 games against the other conference (single game either home or away). Regular season is 9 months long = 39.5 weeks and playoffs are another month (4.5 weeks). No reason that can't work with a slightly larger roster.