Rim size doesn't mean a whole lot, but the hub might be larger/bigger around on the one with 9in rim.
Also FYI, your machine is basically an ATC200S before they badged it that, the 185 from that time was the "S" model, but both machines share a ton of
parts including the wire harness (very rare for Honda to do that). As far as the orig question, front shocks help, but you'll still get the heavy bumps in the back. The back kind of just slams down and bounces around a bit when pushing speed vs terrain. I rode the snot out of a8 84 200es big red before I switched to a 250es and fell in love with the suspension. There's no direct full suspension machine that's the same basic machine as the 200s, closest I'd say is the 200x. If you don't mind switching to shaft drive, the 250sx is a great machine too, or go big and target a 250r or 350x, but they are more speed machines, putting around is more 250sx like. FYI, 200x, 250r, and 350x all have a manual clutch, the 250sx drives just like the atc200, but is shaft drive and quite a lot heavier. I guess what I'm saying is, maybe replacing or adding to your collection is the best solution. Just another option to take into account.
Anyway, steering stems on the 200cc models (except the 200x) from my understanding are more or less interchangeable, but generally you run the front hub and such from whatever machine the front end came from. If you can score a
parts machine with the right front end design you want, that might be the best option cost vs benefit wise. The 185/185S got suspended front end too in the later years I'm pretty sure. I think this is why the auto X is such a popular build, it's basically a fully suspended 200s (semi auto)
Also should note, rim bolt patterns might have changed between models, might want to check that before committing on a front end.
Another thing to point out if you haven't already figured it out, the ATC tires these machines came with were soft and thin so the suspension was the tires. What tire you run and the PSI makes a huge difference on ride comfort and handling. I personally like to keep tires aired up well, so I choose machines that have suspension to take the beating instead of my body. The 200es I started on had the OEM tires, and it rode pretty well, once the dry rot finally took them out, I swapped hubs from the ATC90 I had and holly crap was those tires soft. It was a bit bouncy on the trails, but soft ride lol, pretty sure they were OEM too, and also dry rotted badly.
Anyway, good luck on what ever route you go to solve your issue.