We propose a gradient-based scheme to solve the fluorescence photoacoustic tomographic (FPAT) problem in a fully nonlinear one-step setting, which aims to reconstruct the map of the absorption coefficient of an exogenous fluorophore from boundary photoacoustic pressure data. Adjoint-based gradient evaluation is presented for the FPAT problem in a frequency-domain photoacoustic equation setting. Numerical validations of the resulting Broyden-Fletcher-Goldfarb-Shanno (BFGS) reconstruction scheme are carried out in two dimensions for full- as well as limited-data test cases, and the results are compared with existing Jacobian-based one-step FPAT reconstructions. The reasonably comparable results of the one-step gradient- and Jacobian-based FPAT reconstruction schemes, coupled with the significant computational savings of the former, potentially set up the one-step gradient-based schemes as an advantageous method of choice for FPAT reconstructions. Further reconstruction studies carried out using quantitative photoacoustic tomography (QPAT)-based chromophore reconstructions as inputs to the FPAT inversions show a robustness of fluorophore absorption coefficient reconstructions to the QPAT-obtained inputs.