How can I shave about 1/8" from a 8'x3/4"x1 1/2" strip of oak?
The tool of choice for this type of shaving operation is a router with a ball bearing edge cutting bit. Since the edge that you attached is 3/4 inch thick you would an edge cutting bit that has at least a 7/8 inch cutting edge.

The picture below shows this type of trimming operation in process.



Note that when you set out to do this for your project you will need a more powerful router than the small single hand sized unit shown in the picture. Shown is a small router known as a laminate and veneer trimming router. Also it can be tricky balancing a larger sized router along the 1 3/8 inch edge of your plywood pieces so use care to be safe. I also suggest that before using a router to spend some time researching online and become familiar with the proper cutting direction for router use and to learn about some jigging possibilities that you can use to help stabilize the router base by clamping on another parallel edge of wood or plywood that is spaced away from the cut piece by more than the width of the router bit. The picture below shows one type of jig clamped in place to provide more support for the router during an edge trimming operation.