For people who don't go to national parks a lot, it's probably cheaper to just get day passes. People who only go to a single park a lot can also get an annual pass for just that park at the fee gate. You can also put a sticker over his mugshot, just don't block the text.

I got a spoonbill one in December to avoid this. You could try asking at an REI or something if they have leftovers that they could punch for December.