A number of open problems hinder our present ability to extract scientific information from data that will be gathered by the near-future gravitational-wave mission LISA. Many of these relate to the modeling, detection, and characterization of signals from binary inspirals with an extreme component-mass ratio of ≲10-4. In this paper, we draw attention to the issue of systematic error in parameter estimation due to the use of fast but approximate waveform models; this is found to be relevant for extreme-mass-ratio inspirals even in the case of waveforms with ≳90% overlap accuracy and moderate (≳30) signal-to-noise ratios. A scheme that uses Gaussian processes to interpolate and marginalize over waveform error is adapted and investigated as a possible precursor solution to this problem. Several new methodological results are obtained, and the viability of the technique is successfully demonstrated on a three-parameter example in the setting of the LISA Data Challenge.