The most simple RFID tags have just an unique serial number and don't need any programming. Passive tags at 3 m are basically feasible with the ISO 15693 13.56 MHz standard, but requiring large reader coils and a respective power amplifier, so the 50 $ budget will be most likely exceeded. But that won't be better with any RFID or EAS (electronic article surveillance) technology at this distance.
The other point is, that RFID can't sense an exact position, if the objects are actually on the kitchen counter top, or e.g. hidden in a drawer. Your specification isn't clear in this regard. (The only clear point is about light beam "unworkable", without telling why). The image recognition solution would be perfect in position detection.