Taking this question from a different perspective. Lets forget the black hole. Black holes run into the problem with having unknown interiors inside their inescapable event horizons with the possibility of a near singularity in size at their center due to no known force being able to prevent their continued collapse, so forget black holes. A neutron star is easier to work with as a theoretical scenario.
Imagine, instead, that we have a very high density Neutron star on the verge of sufficient mass and density to collapse. The Neutron star equivalent of the Chandrasekhar limit, aka the Tolman–Oppenheimer–Volkoff limit. This isn't a black hole, but it's as close as you can get to a black hole without being a black hole, with an escape velocity approaching to the speed of light.
Now lets take your scenario:
Suppose two SMBHs (supermassive black holes) are brought close (e.g.
during a galactic merger). Suppose that a "normal" black hole drifts
into the space between the SMBH
I like your scenario because galaxies and their supermassive central black holes do merge and as a result it's theoretically possible that this could happen. Keep in mind, everything in space is in free-fall and objects in free fall don't experience the gravity of the object they orbit. What they experience are tidal forces and the tidal forces of super-massive black holes is comparatively pretty weak. You get the highest tidal forces (just outside the event horizon) from the smallest stellar black holes not supermassive ones. See here or here or here. A Neutron star would withstand any tidal forces from two colliding supermassive black holes with ease.
What would happen is some tidal stretching like Europa experiences from Io and Ganymede as it orbits Jupiter. In the case of Io, this is often called tidal heating, but the heat comes from stretching and squashing and it's continuously generated every orbit due to the 3 body dynamics. Super-dense objects like Neutron stars are much more resistant to stretching and squashing, so the gravitational tidal forces would need to be enormous to have any effect.
3 body tidal stretching outside the event horizon of super-massive black holes is comparatively weak. Much too weak to pull apart a neutron star. You'd get the strongest theoretical tidal stretching from stellar mass black holes and the strongest tidal forces possible from the smallest stellar mass black holes.
For this theoretical scenario to work, you'd need two stellar mass black holes about as small as black holes can form, which would have maximum density in relation to the volume of their event horizon and a Neutron star to fall perfectly in between them, nearly touching them as it falls between them and in such a scenario, the tidal forces might pull apart the high density Neutron star . . . if it was perfectly timed. The improbability of a 3 body scenario like this actually happening is essentially zero.
I should add, I'm not sure how the relativistic factors would affect the tidal forces. Weird things happen with relativity, for example, the escape velocity becomes lower than the orbital velocity and precisely how the relativistic effects would affect orbital velocity and tidal stretching, if they would affect it at all, I'm not sure.
Pulling apart a black hole raises a whole set of impossibilities and to my understanding, covered by @Peterh answer. Pulling apart a dense Neutron star between two small, close to maximum density stellar mass black holes if the Neutron star threads the needle between them as they tightly orbit each other, just barely, maybe, but as noted, it's a prohibitively improbable scenario.
That's not the question you asked, but I think it touches on the idea behind your question.