To answer the question, below are scatter plots of the total precipitation versus number of days reporting thunder, for June and July. Indeed we see that in both months, the highest rainfall totals are not associated with a high frequency of thunderstorms. Instead, the highest thunderstorm frequency tends to be associated with moderately, but not excessively, above-normal precipitation. In June it is possible to have quite high thunderstorm frequency with well below-normal precipitation, but it seems that in July the very driest months tend to have reduced thunderstorm activity.
In conclusion, the wettest summer extremes in Fairbanks are not associated with much-enhanced thunderstorm activity, but in fact quite the reverse: thunderstorms are less common in the weather patterns that bring the wettest conditions. This is evident also in the history of heaviest daily rainfall amounts: of the top 20 one-day rainfall events in June and July (for years with hourly weather reports available), only 5 days reported thunder.