It's pretty easy to use a router with a split fence to plane material. You put a bit of thick material between the outfeed fence and line a tall bit up with the outfeed fence. Press the wood forward across the bit and it will pick up at the slightly raised side. Shift your pressure against the fence once the workpiece starts on the outfeed side.
For the middle piece, flatten one side, then run that flat side against your table saw fence and trim the second edge. If you don't have a Wixey digital angle finder, be sure you get one so your blade is exactly 90 degrees to the table. Pix below.
I agree on using a spline to attach them. But make certain all three pieces are face down when you cut the spline. That will greatly increase the likelihood the top will be flat.
If the edges are not straight once assembled, you might consider using a straight edge and a circular saw to cut one edge very straight, then flush that against your fence to cut the other side. Or you can use the circular saw on both edges if your table saw isn't wide enough.
Note that spline material should run cross grain to the long boards, so you are likely to have to piece it all together. I would buy a slot cutter the width of the thickness of 1/4 ply and use the ply for a spline.
Hope this helps. I found a video on using the router as a jointer. Pretty simple. Some people use playing cards as spacers, but you can also find 1/16 inch thick spacers used to level drywall. I use padding material backing called chip board, same stuff as on the back of tablets.
