Belvedere is a nice track, but it's not close to Arcadia. It's about a 2 1/2 hour drive with no slow traffic. There's just not a good way to get there from Arcadia. Also if you were going to go to Charlevoix, I'd just say go another 25 minutes and play Bay Harbor, which is probably the second best golf "experience" in Michigan behind Arcadia (Public courses anyways). Or save yourself an hour and a half of driving and just play Lochenheath in Traverse City.
Crystal Downs is the best course in Michigan, and rivals anything else in the country (it's routinely listed in the top 10 tracks in the nation. However, it's impossible to play unless you know a member. It's pretty much as private as private gets in Michigan.
Kingsley Club is also a great one, and a little easier to access but still private.
Other courses of note in the area:
Manistee National - Cutter's Ridge is a pretty nice course that's on your way to Arcadia. A lot of people will pair this with Arcadia and play 36.
Crystal Mountain has free golf this year from 12-3. I forget the details, but I believe it's M-Th, 12-3, on their Mountain Ridge course which is a decent track. A little hike from Arcadia, but not too bad (45 minutes to an hour).
Heathlands in Onekema is an okay track.
It should be noted that none of these courses even remotely begin to compare to Arcadia, but are solid tracks at reasonable rates that can be played along with AB. Cutter's Ridge is definitely the best of the group.