Addition: You could tape the cross to a piece of clear plastic and use the table with a bearing guided bit. You could at least see what you're doing.
As to cutting the half lap, consider making yourself an adjustable, exact width dado sled. You set the gap by sliding the Cross piece into the opening, You'd need to very carefull align the slot 90 to the side rails or your cross would be crooked--not acceptable.
When you build the jig, use an engineer's square to position the fixed cross piece at a precise 90. Technically not necessary, but you'll be happy you took the time and trouble later when you use one of the short pieces like a T square to align the jig. Build it nice and strong, you'll find uses for it for decades.
To assure alignment is 90, cut a piece to, say 8 inches long, cut it down the middle and with these pieces against a short side rail on the jug, push the cross up against the short pieces. It will assure your pieces are square to the jig--if the jig side pieces are square (90) to the slot.
Make a nice big jig. Clamp the jig in place, set the width, then tape the cross pieces down.
I always use a bottom bearing mortising bit for this kind of joint because it makes a very flat bottom cut. Here are pix of the sled (Lots of plans and video online), and a mortising bit.
Because you're using an exact width jig, the fit will be perfect, you just need to be certain your bit height is just right. You have to set bit depth of cut to allow for the additional thickness of the jig pieces, plus exactly half the thickness of the workpiece.
