If you want to denoise the signal, it might be best to use the modern-day ENSO data such as SOI and SST, as those have monthly resolution.
With the UEP ENSO proxy, all that is provided is yearly data so that it is difficult to tell how much of the noise fluctuates within a year and whether it is visible wrt to the sampling rate. That seems to be a recipe to generate noise that can't easily be eliminated because the spectral composition is folded on itself, see
You may have better ideas about this than I do. Look at the residual and see if the frequency spectrum is flat? That would indicate white noise.