The localization of directional RF emitters presents significant challenges for electronic warfare applications. Traditional localization methods, designed for omnidirectional emitters, experience degraded performance when applied to directional sources due to pronounced received signal strength (RSS) modulations introduced by directive beampatterns. This paper presents a robust direct position determination (DPD) approach that jointly estimates emitter position and beampattern parameters by incorporating RSS modulation from both path attenuation and directional gain alongside angle of arrival (AOA) and time difference of arrival (TDOA) information. To address the computational challenge of joint optimization over position and beampattern parameters, we develop an alternating maximization algorithm that decomposes the four-dimensional search into efficient iterative two-dimensional optimizations using a generalized beampattern model. Cramer-Rao Lower Bound (CRLB) analysis establishes theoretical performance limits, and numerical simulations demonstrate substantial improvements over conventional methods. At -10 dB SNR, the proposed approach achieves 49% to 61% error reduction compared to AOA-TDOA baselines, with performance approaching the CRLB above -10 dB. The algorithm converges rapidly, requiring 3 to 4 iterations on average, and exhibits robustness to beampattern model mismatch. A contrast-expanded half-power uncertainty metric is introduced to quantify localization confidence, revealing that the proposed method produces concentrated unimodal likelihood surfaces while conventional approaches generate spurious peaks. Sensitivity analysis demonstrates that optimal performance occurs when receivers are positioned at beampattern main lobe edges where RSS gradients are maximized.