Robert, like many problems there are several possible solutions. Your solution offers no advantage I can see, and being more complex has more places to fail. And you will need still more complexity to charge the battery, or do it manually and risk needing it precisely when you've got it out of the circuit. But with a bit of fiddling, it can certainly be made to work.
Don's solution is simple, more robust, and has the added advantage of trickle charging the battery.
Also, both circuits are missing decoupling caps for the regulators (presumably for simplicity in the drawing).
What sort of battery? Sealed lead acid / gel cell I presume?