With these comments, I am trying to put the nail in the coffin (edit: that sounds bad, I mean stick a pin in it) on the ENSO model
One result I noticed from the independent low and high model fits, is that a slight temporal shift exists between the two profiles. This is best observed as a ~1.5 month difference in the Mathieu modulations (i.e. the LHS modulation of the DiffEq):
This translates to the same shift in the forcing factors (i.e. the RHS of the DiffEq). So if I translate each of the factor curves by either 1 or 2 months, this is the alignment I get:
Note that the RED curve is the high interval and the BLUE is the low interval and so the stronger the blue, the greater degree the two agree, since the red will hide behind the blue. The correlation coefficients are shown as well.
So consider that this is the result of about two hours of the Excel Solver grinding away on an optimal solution for each interval. There was no overlap between the two intervals, yet we still get this stunning a result. The 1.5 month discrepancy may essentially be the uncertainty in the collection of the ENSO data. I don't think it's significant as this could be simply a local minimum that the Solver converged to due to inherent noise in the data.
Because it takes a while to run these trials, I haven't tried too many other sets of input parameters. Yet if I take some arbitrary values to substitute for the ( 6.41, 14.6, 4.085, 18.6, 9.3 ) set, the resulting fit is horrible and there is absolutely no correlation in the Low vs High output factors. There may in fact be another set that works as well, but the periods would have to be physically significant to make any sense in terms of the periodic geophysical forcing mechanisms at work.
The remaining two factors that I can add are derived second-order tidal factors corresponding to 5.643 and 3.447 years which are associated with the fortnightly long period tides stemming from nonlinear multiplicative interactions of the nodal, anomalistic, and tropical monthly periods. These are critical for achieving highly precise tidal predictions and so would think they would be relevant here as well. But I have to be careful of overfitting at this stage. These would require twice as long an interval for convergence and so I would lose the low vs high training validation.