I think $250 for a ar type set up with a two stage trigger is a pretty tall order and kind of unnecessary expense for practice. Honestly I think you've got the right idea with using peep style sights for practice and Rapier's suggestion of the Tech sights is right on.
If money is a concern which it almost always is then I say don't worry about getting something similar to an AR in ergonomics or trigger. Just worry about getting some decent trigger time practicing the basics. I'm talking about practicing good sight alignment, sight picture, trigger control, and breathing on each and every shot. You may want to practice getting into good shooting positions with a sling too if your thinking about shooting in matches.
I say just get a decent accurate .22 with a nice trigger and shoot the hell out of it. The 10/22 is a good suggestion and the sr 22 I posted is just a 10/22 with all the AR stuff already installed. You could also go for a Remington 597, they are good too and I think I seen a special in the Sunday adds where Dick's or maybe Bass Pro was selling them for $99 this week. That's pretty hard to beat. Also like other's have noted with these two you can always upgrade as you go and money comes in.
I personally prefer the Marlin Model 60's for a great semi .22lr plinker. They are generally accurate right out of the box and need no aftermarket anything. That's a good thing too because there is not much aftermarket available for it either. That may say something considering they have sold over 11 million of them since 1960(twice as much as the 10/22). Throw some tech sights on top and I think you will be surprised with how accurate you can be with it.