if I disconnect the RC522 3.3V line the LED goes dim (taking power via the SCK, MISO, etc pins) and goes bright when I reconnect the 3.3V
I assume you connect the RC522 3.3V to the ESP8266 3.3V ? are you powering anything else from the ESP8266
are you sure your RC522 is a 3.3V version - it appears there are 5V versions - see using-an-rfid-module-with-an-esp8266
otherwise you may have poor connections (jumper wires are an endless source of problems) or even a faulty RC522