Rather than chance giving you some bad info for this scenario, I did some more digging in my maps and documents. I've been going to the BB area and exploring from west of Presidio all the way to Black Gap Wildlife Management Area since 1973 and have collected a lot of info and documentation. I also did some Google earth viewing. I often use Google earth satellite images for pre and post trip research.
I would disregard my comments or suggestions about the loop 2-track off of hwy 170 that I mentioned. Since I haven't done any of Tapado and only part of Oso, I thought I'd review this. I've driven by Tapado on 170 many times, and Oso does indeed dump into Tapado. The 2-track I mentioned does indeed go closely to Tapado on the west side, but there is a huge ridge that runs the whole way north to where you'd have to hike a very long way before you could "probably" drop into Tapado. Not worth the effort from that direction IMO.
I think your best bet is to go directly up canyon on Tapado on foot to a place you find suitable for your goal. I have often looked at Tapado as I'm driving past it on 170 and been tempted by the neat looking aspect of the terrain there for off roading. There is definitely an old road bed of a 2-track that is patchy, and honestly I think I can see that there is some evidence that 4WD vehicles have driven in there somewhat recently and every so often. It's probably illegal because it still is park property there, I believe, and they don't authorize driving on non-authorized roads/trails. But I doubt anyone would notice from 170.
However, that track in looks pretty challenging very quickly even for a 4WD and probably impassable not far in. The walking up the dry creek bed looks quite doable, however. I don't know how far you'll go, but if you push in enough you'll hit the intersection of Oso Canyon. Oso is actually the larger one, so I'm not sure how Tapado got the nod for being named as the canyon at hwy 170...but it is what it is. Oso will be the bigger drainage and dry stream bed on the right. Further up you might even run into a trickle of water in Oso. This is where I saw cottonwoods and vegetation. I was able to glimpse Tapado from the Guale overlook, but I didn't make it to the intersection. From looking at Google, apparently I wasn't that far from the intersection.
You should have no trouble walking up Tapado. I estimate it to be about 2.5-3 miles to the intersection with Oso. The real elevation climb doesn't really kick in until that intersection. Let us know how it goes.