I am answering my own question here -- I doubt that any of the models that claim ENSO follows a red noise pattern [1,2,3] are justified. As long as the source of the forcing is known and that the response can be characterized, as I have shown above, the stochastic nature is essentially eliminated as a candidate to explain ENSO variability. This follows the typical pattern of whatever is not understood is characterized as noise, but since we understand the forcing and response, the result is not noise, but signal. For example, that hum you hear on your speakers is not noise if you can trace it to the 60 Hz AC power source and subsequent amplification.
However apart from noise, there is a possibility that I have not experimented with -- and that is the strong likelihood that two different ENSO dipole mechanisms may be at play. The canonical ENSO is the one associated with the eastern Pacific, while another one is associated with the central Pacific (see ). In modeling the SOI, I have assumed that a single canonical ENSO was the primary mechanism at work and so solved a single wave equation.
I think that the best way to work this is to find another dipole that follows the second ENSO mechanism, while maintaining the original model for the canonical SOI dipole. Then once these are independently modeled, the two can be "mixed" to see if they improve both dipole model fits. One of the other SST NINO indices may be a potential complementary candidate, but upon examination they all seem to correlate to the SOI index rather well (or more precisely anti-correlate since SOI has the opposite sign to temperature) . This means that it will be hard to isolate the second dipole.
 D. L. Rudnick and R. E. Davis, “Red noise and regime shifts,” Deep Sea Research Part I: Oceanographic Research Papers, vol. 50, no. 6, pp. 691–699, 2003.
 “REDFIT: estimating red-noise spectra directly from unevenly spaced paleoclimatic time series$.” [Online]. Available: http://www.manfredmudelsee.com/publ/pdf/redfit.pdf. [Accessed: 01-Dec-2012].
[3 ] M. Newman, S. Shin, and M. A. Alexander, “Natural variation in ENSO flavors,” Geophysical Research Letters, vol. 38, no. 14, 2011. http://onlinelibrary.wiley.com/doi/10.1029/2011GL047658/full