I had my 6ft BRB in a tank like that when I first got him, it was just bigger. He is a great climber and a big time burrower but he was never able to get out of the tank. The screen lid was not held down with anything other than the built in tabs yet he was never able to escape. I also had the background mounted to the glass with an epoxy and never had an issue there either. A BRB is much stronger than a ball and your cage should work fine if you just mounted the background to the wall but leaving it on the outside is certainly fine too.

Trust the tank. Its made for keeping critters in and it should be great as long as you latch the door and don't break the tabs on top. Covering part of the screen is just part of having a screen lid. Aluminum foil tape can work well for that too.

Also a tip. The front of the cage has a vent that runs the entire front. You can use electrical tape and run it along the vent to cover it(on the outside) or a portion of it. This will help keep your humidity in too. The electrical tape is the same width as the vent so it disappears when added to it nice and straight.