Could the black hole consume the radiation? If so, could the pulsar radiation absorbed by the black hole make it grow steadily more massive over the course of billions or trillions of years?
Having the pulsar emiting radiation without decaying, the black hole not decaying via Hawking radiation and also the orbits of both objects not decaying and in a particular dynamic, I would say yes, veeeery slowly.
Thing is the radiation from the pulsar comes from particles that fall in it (dust, etc) and are "reoriented and shot" (roughly speaking) as high-energy radiation near the poles because the magnetic field. The black hole would attract even more particles than the pulsar so considering both objects in a space with same density of particles, the black hole by itself should suck in more matter/energy compared with matter/energy from the pulsar beam.
Also consider orbital implications. A black hole (unless supermassive at center of galaxies) and pulsars have equivalent masses. Not the same, but not different in orders of magnitude if the black hole is a stellar remnant as the pulsar. That means that the orbit will be, probably, more like a double-star system, with the black hole moving also around the center of masses (you can thing about it as Pluto-Charon orbit system). Now. Consider how stars are formed (from a dust cloud, it starts to collapse, gain angular momentum, etc). Chances are that the orbits and the spins of both objects are in a very similar planes. Think that the beams of the pulsar are shot mostly from the poles (with some inclination though). I think is almost impossible for a naturally formed stars that one becomes pulsar, the other black hole, and also they orbit in a way that the beams of the pulsar are going to feed the black hole. The beams would just scape from the system, forming ~90º relative to the line from the pulsar-black hole.
Last, about the "feeding". Lets consider that all the above is possible, just very unlikely. We have a scenario where the orbit of a pulsar is ~90º compared with its spin, like Sun-Uranus. Then a fraction of the time it feeds the black hole (twice a "year"). Black holes eat stars sometimes. The size of the black hole grows when it eats, and is one of the theories about the mass of supermassive black holes in the center of most galaxies (just eating... the other is direct formation via a sudden collapse of super-massive dust cloud even avoiding a star formation). I try o point that this feeding via a pulsar beam would be super slow compared with direct feeding of surroinding matter like the supermassive black holes do.
I would say to the black hole: Sir, eat the pulsar :)
And maybe it would become true: considering very long periods of time, orbits decay. The black hole would maybe collide/absorb the pulsar (as recently detected via gravitational waves) Finally, depending on the period of time we are talking about, we have to consider black hole evaporation throught Hawking ratiation (it would make it smaller). On the other hand, I considered pulsars powered by matter orbiting/falling in it. there other types, so a rotation-powered one, after trillions of years, could maybe decay in an almost neutron star. I don't know about the numbers so maybe these effects are almost zero, not sure.
PD. This is my first answer in StackExchange and I'm not a native english speaker, feedback appreciated ;)