This article describes a source that seems to start and stop at regular intervals of several days (much longer than any known pulsar's rotational period) but within the active intervals, the bursts appear to be random. That suggests a more complex mechanism is at work.
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So they just go around and around, including the light, from the point of view of a far away perspective
As a method to communicate with people you don’t know exist, it’s not very good because chances are they won’t be viewing your satellites edge on. For people you know are there, it’s needlessly complicated and expensive.
If you’re going to go to the trouble, you’d be better off making a Dyson swarm and cloaking your star completely. You’d get energy back and be just as detectable for a civilization that thought to compare visible and IR sources (which some Earth scientists have done, with negative results).
The only exception to this is the Crab pulsar, which is observed to occasionally have "giant pulses" which are 1000s of times brighter than a typical pulse. If you put the a somewhat brighter version of the Crab pulsar in an FRB host galaxy at the low end of the known distances, you might just about be able to see it.