A bottom board that is built into the hive stand could work, but I think the chances that it will work well are not that promising.
For one thing, assuming you want a consistent bee space between the bottom of the frames and the screen, you may find that more difficult to achieve than expected. The only way to achieve a consistent space is to ONLY use a specific combination of frames & boxes from a given manufacturer. There is a wide variance in dimensions between frames/boxes when looking at different manufacturers. See this article for details:
We made something similar.
We attached two boards to the bottom of the sides of the screen bottom boards.
The boards are as long as the bottom board and wider then the thickness of the hive walls.
They are attached so they project inward to hold mite count boards.
The mite count boards can be inserted through the front slightly below the entrance.
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