You can increase the strength of the mitered joint by applying glue to the end grain, letting it dry, than glue up again when you insert the spline and assemble the box. If you don't want the spline to be visible on the top or bottom, you'd need to make a stopped groove and insert the spline in that. It will show when you open the box unless you get mighty careful to stop the spline groove so it won't show at all.
This method requires careful measuring and stops setup on a router table, but has the advantage of keeping the box's exterior match. If it were my project I'd allow the spline to be visible on the inside, but not the outside. All this becomes moot if the box is really shallow. Anything over about 4 inches the stopped spline would be an option.
I use 1/8th thick purpleheart for most of my splines in picture frames because I like the contrast. But using the same wood will make the spline less noticeable.
The more I do, the less I accomplish.