Estimating entropy measures under progressive censoring poses a significant challenge in reliability and lifetime analysis. Despite the widespread use of the inverse Gaussian distribution in modeling skewed lifetime data, a comprehensive inferential framework for its Shannon and Rényi entropies under progressive Type-II censoring remains absent. This paper develops a unified Bayesian framework integrating maximum likelihood and Bayesian inference under squared error, general entropy, and LINEX loss functions, employing Lindley’s approximation, importance sampling, and Markov chain Monte Carlo methods. Two parameter configurations were examined to assess robustness under varying likelihood surface complexity, along with prior sensitivity analysis. Results demonstrate that Markov chain Monte Carlo and importance sampling are the only consistently reliable methods across all scenarios, whereas maximum likelihood suffered severe bias and collapse of Wald interval coverage under challenging settings, and Lindley’s approximation exhibited numerical instability at small samples. Shannon entropy proved substantially more sensitive to parameter variation than Rényi entropy, with the LINEX and general entropy loss functions showing superior performance. The study offers clear practical guidance, validated on real lifetime data.