Measuring proton-proton interaction cross-sections at center-of-mass energies above 40 TeV remains a significant challenge in particle physics. The Pierre Auger Observatory provides a unique opportunity to study the interactions at the highest energies through the distribution of the depth of maximum shower development ($X_\mathrm{max}$) observed by its Fluorescence Detector. In previous studies, the determination of the interaction cross-section at ultrahigh energies has relied on the assumption that the tail of the $X_\mathrm{max}$ distribution is proton-dominated, which restricts the analysis to a limited energy range below the ankle and introduces related systematic uncertainties. In this contribution, we adopt a novel method for the simultaneous estimation of the proton-proton interaction cross-section and the primary cosmic-ray mass composition using data from the Pierre Auger Observatory, avoiding assumptions about one quantity to infer the other and thus improving the accuracy and robustness of our analysis. In addition, a systematic shift in the $X_\mathrm{max}$ scale is fitted to account for both experimental uncertainties and theoretical constraints on the modeling of particle interactions. The obtained results are consistent with previous analyses and provide additional constraints on hadronic interaction models. The measured proton-proton inelastic cross-section at ultra-high energies agrees well with extrapolations of accelerator data. The inferred cosmic-ray composition and the $X_\mathrm{max}$-scale shift are also compatible with previous estimates.