I'd need to be able to have it ignore a previous tag rather than read multiples
It's the same thing, if a tag is in the field it is interfering with the field and unless it is smart enough to back off and not transmit you can't read anything else.
I want to do a similar project where I have a physical sequencer using RFID tags. I could do this with 32 readers but that is a bit silly. An idea I have had is to use one energising coil over the whole area and multiple pickup coils in specific locations. Then multiplex the coils. In that way each tag will in effect have its own reader but I only need one actual reader. I hope to work on this over the summer break, so you may be able to adapt that idea (if it works) to your project.