Estimating influence on social media networks is an important practical and theoretical problem, especially because this new medium is widely exploited as a platform for disinformation and propaganda. This paper introduces a novel approach to influence estimation on social media networks and applies it to the real-world problem of characterizing active influence operations on Twitter during the 2017 French presidential elections. The new influence estimation approach attributes impact by accounting for narrative propagation over the network using a network causal inference framework applied to data arising from graph sampling and filtering. This causal framework infers the difference in outcome as a function of exposure, in contrast to existing approaches that attribute impact to activity volume or topological features, which do not explicitly measure nor necessarily indicate actual network influence. Cramér-Rao estimation bounds are derived for parameter estimation as a step in the causal analysis, and used to achieve geometrical insight on the causal inference problem. The ability to infer high causal influence is demonstrated on real-world social media accounts that are later independently confirmed to be either directly affiliated or correlated with foreign influence operations using evidence supplied by the U.S. Congress and journalistic reports.