Hypothetical question: when a pair of identical stellar black holes are very close to one another wouldn't the side facing each other experience lower gravitational force? How is this logic un/sound?
Your logic isn't unsound. By symmetry there would be a gravitational neutral point between them.
Of course, the black holes could not then be described by the Schwarzschild metric, since that only applies to a spherically symmetric situation.
As you brought the black holes closer together, the isopotentials would become distinctly aspherical, the event horizons would be complicated shapes and ultimately would merge - although as the black holes would presumably be in orbit and rotating, the spacetime geometry would be even more complex than that.
What would really happen is that the 2 black holes would spiral around and into each other so fast that any theoretical safe point wouldn't be safe for long, like a few millionths of a second at most. But if you imagine a snapshot in time, where 2 black holes are near each other and your equidistant from both of them, that would be a gravitationally neutral point - as Rob Jeffries says.
The experience of zero gravity happens in orbit anyway, so perhaps a better way to look at is is, is whether you'd experience gravitational curvature of space, where your direction is pulled off in the direction of the massive object, and if you're between 2 massive objects and you experience an equal tug from both, you would in a sense, fly in a straight line between them.
You would still experience tidal effects though and the tidal effects near a stellar mass black hole would be sufficient to tear you apart. The Neutral point is exactly that - a point, so a 3D ship would inevitably feel more tug from one on one side and more tug from the other on the other side, as well as being drawn towards the point on approach and experience a tug back to the point on exit.
If it's 2 super-massive black holes then the tidal effects are less of a problem and that might be a more interesting and longer lasting journey, but the gravitationally Neutral part is still just a point, but the "drop off" away from the point would be less intense.
Now consider if the Schwarzchild radii of the 2 black holes were touching - in our thought experiment, that's a different story and the gravitational sweet spot is now inside a bigger black hole, not in a safe spot between 2 black holes.
Think about what happens when you have a 3 stellar mass black hole, radius 8.8 KM and another 3 stellar mass black hole, 8.8 KM, and the Schwarzchild radii are just touching, the singularities are 17.6 KM apart in that instant, but you now have enough mass for a 17.6 KM Schwarzchild radius black hole, so in effect, it's now 1 bigger, not 2 smaller black holes and the zero gravity sweet spot, is right in the middle of the 17.6 KM radius black hole, Because as you double the mass, you double the size. If the black holes aren't touching, then it's theoretically possible to fly through there, but the instant they touch, both black holes and everything in between them is inside a bigger black hole.
This, of-course, an oversimplified model assuming stationary black holes. In reality, I think you'd get a much more complex gravitational dance.
But the answer to your question "can they cancel each other" - no. They can only add to each other. Gravity between 2 objects can have a balancing out effect, but never a cancelling out effect. Think of the 2D representation of gravity
as you push 2 black holes closer together, there's never a point where the holes cancel each other out and create a zero gravity between them, but there is a ledge of sorts between them, below the near zero gravity of empty space. As the holes approach, they make each other bigger. The balancing point between them, is still inside the gravity well and as they get closer, the sweet spot between gets deeper and deeper in the 2-d representation.