I would veneer the fronts up to the flats, and then veneer on the flats at the ends. By following this process, this will allow the veneer to be clamped in place and resist lifting. You may need to create a back bevel on the flats to help make the joint seamless. Depending on how thick the veneer you are using, you can possibly hammer veneer it with hot hide glue, or caul it with bending plywood and old brown glue.
FR