Mystery Engine, need your help!
Well, the one piece cylinder/head means it's not easy to get a measure of the top of the cylinder bore but measuring the top of the piston is close enough for gov't work. This is your bore.
Then you measure the stroke. It's easier on half crank engines were you can slide the con rod off the crank journal. I'm guessing your engine has a full crank with counterweights on both sides of the rod journal. This makes it a little tougher.
If the spark plug isn't angled too low, you might be able to use a small 'indicator' like a piece of wood or plastic. Squirt tubes from wd-40 works okay. Place the indicator down the cylider touching the top of the piston. Mark it at TDC then BDC and measure the difference. This is your stroke.
Multiply .7854 x bore x bore x stroke and this is your displacement.