The strong requirement that an annual impulse interacts with the tidal forcing suggests that the [18-year Saros cycle of eclipse](https://eclipse.gsfc.nasa.gov/SEsaros/SEsaros.html) occurrences is operable for ENSO modeling, as it is a longitudinally *regional* (i.e. Pacific ocean) cycle. In contrast, the entire earth reacts to *global* tidal forces, which explains why the modulation in LOD is closer to the 18.6 year nodal cycle, which ignores the regional eclipse cycle.
Since we are only using the annual cycle along with the 3 primary lunar monthly cycles in modeling ENSO, one can see how there may be an 18 year cycle observable in the model, given this from eclipse.gsfc.nasa.gov:
> "One Saros is equal to 223 synodic months. However, 239 anomalistic months and 242 draconic months are also equal to this same period (to within a couple hours)!
> 223 Synodic Months = 6585.3223 days = 6585d 07h 43m
> 239 Anomalistic Months = 6585.5375 days = 6585d 12h 54m
> 242 Draconic Months = 6585.3575 days = 6585d 08h 35m
The chart below is a typical tidal model fit to ENSO data with a given LTE modulation and daily amplitude spectrum. Below that in the middle panel is the baseline tidal cycle plotted at a sub-monthly scale, overlapped with the time-series shifted by close to 9 years, which is half the Saros cycle (equivalent to a solar-eclipse-to-lunar-eclipse alternation cycle). The repeat pattern is readily apparent here and points to how much of the *information content* is likely found in a single 9-year interval (horizontal dotted arrow). Hint: one actually has to look closely to observe the slight discrepancies between the two.
I experimented with an independent model fit that iteratively evolved with a slightly different tidal factor pattern as well as a different set of LTE modulation levels. As the LTE modulation can only create harmonics of the original forcing, it's perhaps not surprising that it takes effort to find the optimal modulation. Yet saying that, the information content of the sub-monthly tidal forcing again shows a clear repeat pattern following the 9-year half-Saros cycle (middle panel). The comparison of the two models frequency spectrum is shown in the lower panel.
The bottom-line is that even though the 9 and thus 18-year period is clearly evident in the model, it is not exact. If it was indeed exact and all 3 of the lunar cycles and annual cycle precisely aligned on that interval, then the ENSO cycle will also repeat on a 9-year interval, since the LTE modulation can't do anything to the fundamental period (i.e. it can only change a waveform from a sinusoid to a misshapen sinusoid with the same period). But since there is a slight drift and weave to the cyclic components and any error will accumulate over time, the fitted ENSO model emerges lacking a clear 9-year repeat modulation.