They only accept the CPW Parks Pass. If Stone Bridge was inside the monument you would have a good argument, but it is a CPW/AHRA site. The America the Beautiful Pass doesn't cover FS or BLM lands operated by another entity ( like ski areas). Browns Canyon is a National Monument, but most access points are not part of the designation.
I would write a letter, including photos of signs, to the AHRA pleading ignorance this time. Perhaps since the designation is relatively new they'll let you off with a warning.
The $7 fee (or CPW pass) helps provide restrooms, parking areas, and Rangers who remove wood, assist others, and literally save lives. I think it's worth it.