We study a finite-time quantum information engine in which a two-level system is measured by a quantum harmonic oscillator acting as a meter and where useful work is extracted conditionally on the measurement outcome. Using multi-objective optimisation, we find a Pareto-optimal trade-off between extractable work and its fluctuations: reducing fluctuations leads to increased costs, e.g., increased information consumption, number of operation cycles or decreased work extraction. In the limit of a highly accurate meter the work distribution, its moments, and the Pareto front can be obtained analytically. In this regime, the work statistics of the engine reduce to those of a qubit in contact with a single thermal bath. We further analyse the associated information flows by examining the mutual information and Fisher information, and show that the Pareto-optimal engine designs lie very close to extrema of the latter with respect to the operation time of the device. Our results provide a compact description of the trade-offs between work and its fluctuations in quantum information engines.