The timing of a single pulsar would not be reliable enough to detect gravitational waves. Instead, each collaboration monitors an array of dozens. As a result,

The timing of a single pulsar would not be reliable enough to detect gravitational waves. Instead, each collaboration monitors an array of dozens. As a result, they have found a signature called the Hellings–Downs curve, which predicts how, in the presence of gravitational waves coming from all possible directions, the correlation between pairs of pulsars varies as a function of their separation in the sky. Not sure it answers your question but my impression is they simulate results of all possible effects and then see which one(s) the data correlates with. So if there are multiple causes that could produce identical effects then I doubt they could distinguish between them.