hardware advise on short wireless comunication (rfid)

hi

i got a projekt which i cant figur out what hardware to use.
i want to build a catdoor that open when the cat walk close to the door 1 meter (3-4 feet).
my first though was to use LF rfdi. but i the range is to short, i found one claiming 10 cm or 4 inch.

what im looking after is something like rfid, a reader of some sort and a water proof tag with a range of at least 0.5 meter (1,5-2 feet).
and a price range of 10 to 60 €

any tip would be to great appreciation

I bought a two channel alarm fob off eBay - they sell them with a receiver and really output for a few ££. They work over several meters and I've one on my cat flap
. Open up the fob and wire your switches across the fob buttons - job done