The depth of the maximum of an air shower development, $X_\text{max}$, as observed with fluorescence telescopes, is among the most sensitive observables for studying the interaction characteristics and primary composition of ultra-high-energy cosmic rays. However, precise measurement of the interaction cross section remains challenging, as standard analyses often rely on assumptions about the composition, which are closely tied to the validity of specific hadronic interaction models. In this work, we discuss a method for the simultaneous estimation of the proton-proton interaction cross section and primary mass composition, addressing the limitations of separate measurements. The inclusion of the $X_\text{max}$ scale into the fit further accounts for systematic uncertainties in the data and theoretical uncertainties in particle production. The performance of the method is evaluated using simulations that include detector responses under realistic conditions and with a particular focus on assessing the systematic uncertainties of the fit.