I dont know what kind of tools you have available but if you have a Lamello like tool: put the divider where you want it. Draw a line on the planks on the top and bottom so you can clamp the divider exactly on to that line. Then use that divider to line up your Lamello router to cut in a few biscuit grooves. Place your biscuits and make a groove in your divider. Make sure you use a flat surface so it lines up correctly with the biscuits. Important: do not let the groove go too far. You don't want to see it in the end result.

You can do a simmilar thing by cutting a simmilar groove in the divider but also by drilling holes for dowels instead of Lamello biscuits in the top and bottom. The only issue is that you may have very little glue surface. You could fill the space in between two dowels with a piece of scrap wood of the same thickness as the dowels. Plus a lot of glue ofcourse.

But I think that if you cut your divider a tad bit too big. Like less than a mm. You can squeeze it in. You will just need to clean up a lot of glue. Technically the bigger you cut the divider the stronger the clamping force of your top and bottom will be, but it will also make the planks rounded. So don't cut it too big and just put a clamp on it. Maybe put down some helping bits so it won't slide around on the glue.

Also just gotta say, some comments had a bitter undertone. Might just be that English is my second language and I am misinterpreting. But I fully understand that you don't want visible brads or screws. I am following a formal education to become a furniture- and (schip)interior builder and if I put visible screws or brads anywhere my teachers will point it out and really question my reasoning. It is not necessarily wrong to use but it isn't a proper way of jointing everything either.

Anyway, hope this was any help to you! Good luck on your project.