This study develops a novel statistical framework that integrates the Nadarajah–Haghighi (NH) lifetime distribution with progressive first-failure censoring governed by a beta-binomial (BB) removal law. The proposed approach provides a flexible means to capture variability in dropout processes and offers more robust survival estimates compared to conventional censoring laws. The NH distribution is particularly appealing due to its flexibility in modeling both monotone and non-monotone hazard rates, making it suitable for complex cancer survival data. The estimation of NH and BB parameters is performed through both classical maximum likelihood and Bayesian methods. For the Bayesian approach, estimators are derived under symmetric squared-error and asymmetric generalized entropy loss functions, and a tailored Metropolis–Hastings algorithm is implemented to approximate the intractable posteriors. Asymptotic and credible interval estimators for each unknown quantity are also obtained, using the Fisher information and Markov chain simulations. These point and interval estimation methods are used to evaluate the reliability and hazard rates of the NH lifespan model. Extensive Monte Carlo simulations are conducted to assess the performance of all estimators across various configurations, including different failure percentages, group sizes, and prior information. Results consistently show that Bayesian estimators, especially those employing informative priors and asymmetric loss functions, outperform the frequentist approaches. The model’s validity is demonstrated with two real-world cancer datasets: leukemia remission times and breast cancer recurrence-free survival, both characterized by progressive censoring, dropout variations, and the influence of radiation treatment. In both cases, incorporating beta-binomial removals provides a more realistic and flexible approach to modeling failure risk and subject attrition, thereby confirming the model’s robustness in complex clinical survival settings where patient heterogeneity often leads to early dropouts and intricate censoring schemes.