Hey all, finally took a ride down to the sheep bridge this afternoon. FR 269 was R-O-U-G-H... 5 mph most of the way. High clearance is definitely required, but I didn't have or need 4WD. There were a couple of sandy spots where I was spinning my tires for a second, but there was no doubt about making it home. Just be patient and take it slow. I could see this road going quickly to impassible after a rain, though.
Took 3 hours one way from Anthem to the bridge via Carefree Hwy -> FR 24 -> FR 269 (68 miles). The entire length of FR 24 was easily car accessible.
For reference, I drive a stock 07 Ranger Sport 2WD 4.0L with A/T tires. The tires did come in handy, and you would need some sturdy ones for all the rocks.