A few words to the wise:
1. Many neighborhoods have restrictive covenants that prohibit erecting a cage in your back yard, even if the local zoning laws permit it (and you might have problems there, too). Check on this BEFORE you spend the big bucks, not after!
2. Thanks to the active imaginations of Williamsport's fleet of lawyers and insurance naysayers, Little Leagues everywhere are prohibiting teams from holding any team events other than sanctioned events at sanctioned times and places. Sometimes it's just because the liability insurance policy won't cover off-premises events. But especially when you take kids to your home, you get all the talk about predators and all that stuff.
What we used to take for granted can now be a violation of rules even if you are as pure as the driven snow. I used to bring pitchers and catchers to my back yard bullpen for extra work, but now they won't let that go on. Some people even whined that, irrespective of insurance and other concerns, it was just an "unfair advantage" for our team! (Since when is it "unfair" for kids to get ahead by putting in extra time for extra practice?)
If you ask me, our generation has fouled everything up for everyone with all this stuff, but you'd best be careful not to get yourself into hot water.
Happily you can still practice extra with your own son! And there's always travel baseball, where Williamsport can't tell you what you can and can't do! Just you and your good judgment and conscience.
P.S. My neighborhood's restrictions won't permit a batting cage, but we do have a pop-up net that we use as a backstop for bullpens, and for batting tee work. When we're done it stores away out of sight. And it only cost $80, not $2,000.