I've programmed a lot of BLE based applications recently, so I can easily create what you're looking for. I can create an app that runs in the background, scanning for other nearby players, alerts the user when another player is near, and provide the gaming portion that would then start. My budget is a bit higher than your request, but I can deliver top quality apps. The first app I developed for iPhone was the highly successful iBeer, with over 65 million downloads. Let me know if you would like to discuss further.
Thanks,
Chris Howell
etherVision, inc.