I entered my revised schematic into the freeware Eagle layout editor. I had the bright idea to put the whole circuit, including loop antenna and battery, on a circuit board the size of a standard ISO 7810 card. This way, the RFID sniffer has exactly the same size as the cards to sniff...
The circuit is powered by a 3V lithium coin cell. These are small and flat enough to be included on the card. I also added a small push button switch, so the circuit is only powered while the switch is pressed. This will guarantee long battery life, since it can't be left powered on by accident.
The first design of the single-sided layout fits nicely in a corner of the card, leaving enough room for a square loop antenna and the battery. Since the layout is single sided, the circuit board can easily be fabricated on the kitchen table, or even silk screened with conductive ink! All parts are surface mounted, so there are no holes to drill. The underside is totally flat and free of electrical circuits.
Total parts cost (excluding board, including battery) is about 5 euros.