OK, an update.
The bike runs great!
Smooth, responsive, doesn't foul plugs, (still ugly as ass) but a pleasure to ride.
Uses oil. About a quart in 300 miles.
If you really wrap the motor (rev it), when you let off the throttle you get a puff of smoke.
Valve guides?
My first mechanical suspicion after its long sit: stuck rings. But, before doing anything about that, check the ignition timing and points gap? If the advancer is stuck (from stiff grease and crud) and not advancing/retracting the spark, it will act a lot like you describe. Also, if the points gap is too small (I'm presuming there is no electronic ignition, here) it will cause a weak spark which will burn oil, too.
For those possibly-stuck rings: pull the plugs and squirt in about 1/2 ounce of Marvel Mystery Oil or Bardahl "frees sticky valves" top end detergent oil, let it sit overnight, then repeat and let it sit 2 days. This will begin to loosen the carbon that makes the rings stick. Most K2 bikes ran a little rich and the pistons carboned up pretty badly. Honda did not slow down the spark curve on the advancers for the first few months of K2 production with those HM341 pipes, and the result was an overly-rich mix. Check your mainjets: if it still has the 110 mains, consider switching to 105 for the time being, until you get the rest of it sorted out. This will reduce the carbon issue. Today's fuels are slower burning than the 1970s gas, so carbon will be a bigger problem with the richer jets. I suggest using Shell mid-grade (or even regular) fuel for now to help clean up the old remains: you can think about premium fuel after some of the old carbon is burned off.
I'm so jealous of that perfect K2 seat: they were the most perfect fit on these bikes, IMO. Someday I'm gonna get me a new one for my K2...
