Yep, a bagger needs a fairing.
If - big if, here, you could find one, back in the '60's, Bates, Buco, Wixom and probably other companies built touring-style, handlebar / fork-mounted fairings for bikes like the CA72 /77 (250cc / 305cc) Honda Benly Dream. It would be about the right size for your 350 - a shrunken version of the one you have.
I have one particularly vivid memory of a Dream, everything painted bright white - frame, tank, fender, bags and Buco fairing; the guy was wearing a matching white riding outfit made from some satiny material and a white Buco helmet to top it off. I think the pants legs and jacket arms had fringing on them, but memory might be playing me false, here (it was over 50 years ago).
If I get time, I'll dig through my old bike books to try to find and scan a couple of the old fairing ads and post them.
It might sound odd in this age of 1000cc commuter bikes, but back in the sixties it was common enough to see guys out touring on 250 and 350 cc bikes. My older brother once rode his CB350 (bags, windshield, carrier, crash bars, etc etc etc) 700 miles in one day and I did a bunch of 400 and 500 mile day rides on my '67 YR1 Yamaha 350)
Love your build.
Pat