I have a pair of Five Ten's Guide Tennies ($70) and use them for much some of the same thing you have in mind. They're a nice shoe, with nubuck leather uppers, grippy Stealth-brand outsoles with a dotted pattern to create more edges for grip, laces that extend to near the toe so you can fine-tune your grip.
I haven't used them as a light backpacking shoe and am not sure how they'd be for that. Ankle support is non-existent, which is an issue for me but may not be for everyone. And sole cushioning is minimal—more like a rock shoe than a hiker. So that could be an issue for extended hiking.
You might also take a look at the La Sportiva Boulder X approach shoes ($95). Similar to the Guide Tennies, with nubuck uppers, sticky outsoles, and tunable lacing. But it also has a layer of EVA (the stuff used in running shoes) in the midsole. That creates a cushioned layer that makes the Boulder a little less good on the rock, but a whole lot better on the trail.
You'll find the same thing in the Scarpa Zens ($129), which may be the most deluxe of the current approach shoes. Suede uppers, with lots of rubber randing around the outside of the shoe for protection at heel and toe. Like the Boulder X shoes, the Zens also have EVA in the midsole for good cushioning. But the Vibram outsoles also offer plenty of stickiness.
Maybe you can find a shop with some sort of climbing wall, so you can walk around the store with different shoes, and try a move or three. The fit on all of these is pretty snug, so you'll want to make sure you find the one that best matches your feet.