I would think one of those Dremel looking saws ("Rotozip"?) would work well. I think there is even a saw blade (looks like a drill bit) for the Dremel.

If you are going to use a saw, I would clamp a thin sheet of plywood against the veneer side, and cut through both the veneer and plywood. That way all of the splinters end up on the plywood and the veneer looks perfect.

You can put a sheet of paper towel between the veneer and plywood to stop the veneer side from scratching. That way you just cut through all of the layers and have a perfect finished piece.

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