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:
http://www.beeculture.com/content/beespacearticle.pdf
In certain combinations, the amount of space between the bottom of the frame and the bottom of the box varies as much as 0.4" (slightly more than 3/8").
While a separate bottom board does not eliminate those variances, it does allow you to more easily swap in a different bottom board to solve the problem.