The birds mouth joints don't offer a smooth transition on the outer
face. They are usually used when the piece will also be shaped via
turning or planing. They are not that strong from an internal pressure
resitance point of view either.

You might consider splines\loose tennons. After you miter the edges of
the slats, rip an 1/8th inch dado down the entire length of each edge
about 3/4" deep. Then make 1/8" x 1 1/4 strips (I would use White Oak)
and glue them in using waterproof glue.

For added strength you could go for wider dados\splines, I just like
1/8" because of the obvious ability to do it with the TS blade. You
could also add screws or dowels from the inside to really lock in the
splines.