[squarefoot]

Shouldn't they be able to discriminate the type of object (single star with orbiting planets that obstruct radio waves, or dual star plus planets where the emitter itself also moves around an orbit) by analyzing the received carrier frequency and see if it's also modulated by a much lower frequency (Doppler effect) compatible with orbital motion?

[teraflop]

FRBs don't really have a "carrier frequency" in that sense. The radio burst occurs a broad range of frequencies, which can vary substantially from one burst to the next (there are graphs of this in the paper linked above). But there's no sharp peak that can be accurately measured, as far as I know.

And even if there was, we don't understand the mechanism behind FRBs, so we wouldn't necessarily be able to distinguish a Doppler shift from some other effect that caused the frequency to change.