Both the crank and con rod are different by 1/4". Yes the bore size is the same at 2.938 inches so I don't know how you can bore your engine out and end up with the same diameter as the larger displacement engine.
You say your bore measures 2 7/8 but that equals 2.875 inches, not the 2.938 inches shown in the Kohler K manual. Perhaps you are not using the correct tool to measure your bore.
Why not stick with the 7 HP engine by purchasing a new con rod and getting the crankshaft polished? i don't think that one more HP is going to make a huge difference for you. Take the bore to 10 thou oversize, do the valves, make sure the head and deck or dead flat. Install new valve guides and springs.
Parts to do this are readily available on e-Bay.