Parksguyy, am I correct in thinking your summer high temperatures get up to around 26c/80f degrees? If so then the screened bottom board won't really be helping in regards to lowering any hot temperatures but only with ventilation (as you stated). By turning the solid bottom board around you will be effectively doubling your ventilation and also creating ventilation at the back of the hive where before there was none. That should be plenty of ventilation for the bees in the summer and will create a spot for your sticky board.
I doubt you have to worry about small hive beetles up that way but you could possibly even slide an oil tray in the bottom if you needed to...you might have to build a small shim to raise the screened bottom board up a bit to allow the tray to slide in. Be careful with the sticky board (and oil tray if you use one) and block off the entrance to it at the back of the hive so the bees can't get stuck in it.
When the mites are dislodged from the bees during cleaning they simply fall to a solid bottom board and then hitch onto another bee that walks by. Simply the fact that the mites will be falling through the screen and not onto the solid bottom board where the bees walk will decrease the mite load on the bees. From what I understand the mites cannot travel too far and have a hard time climbing back up if they fall through a screened bottom board. If your bottom boards are similar to mine, the mites will fall roughly 5cm from the screen to the solid bottom board. They will then have to crawl to the edge before they can crawl back up the side...I don't think they are mobile enough to do that in most situations. With a sticky board they're not going anywhere.
I think it's a good move, but...I am a 1st year beek.