This turned out to be a recon trip. I expected problems with the 4x4 road but I didn't anticipate problems with finding it. Here's what I learned: FR44 (the eastern intersection with the Bloody Basin Rd) is ~18mi from I-17 and there actually is a small "44" sign identifying it.
Unfortunately I sampled two different unlabeled dirt roads a mile or two before FR44. This cost me time and in the end I walked from ~1mi up the second one, though I soon realized I should be in the next valley to the east. I toured the high ridge (west of FR44) and explored the drainage just east of Mesa Butte before following it and a smaller tributary to FR44.
I followed the road to Bishop Creek and then went east in the drainage toward Turret Peak. The walking is not bad, a little branch ducking and rock hopping. I only went as far as the 2nd crossing of the jeep road and walked the road back to FR44/Bishop Creek. The drainage and road are probably about the same timewise.
Walking out FR44 I decided it didn't look too bad...probably doable in stock SUVs, though I didn't actually follow it all the way back to Bloody Basin Rd. I angled west up the ridge to get back to my car.
I was back early enough to take the long dirt road, past Seven Springs, back to the Cave Creek area. It's a real thrill when you crest the ridge overlooking Verde Valley and think (for a moment) you're going right off the cliff..