RFID help

I have a $300 Dog door that uses ultra sonic collars to open it. I want to convert it to a passive RFID (UHF?) where if the dog with the RFID Tag gets within 12 inches of the door it will open and as soon as the tag loses range it will close. I am trying to find economical solutions not a $400 RFID Reader. I know there are modules that read 4 inches. It's been done before, however, I am not ready to spend that much money especially since I have an EE degree. Here is the site where it's been done. Smart Pendant RFID Control

Any suggestions of a $100 range solution or less?

Thanks

Did you get anywhere with this?