Small-size acoustic arrays exploit spatial diversity to achieve capabilities beyond those of single-element devices, with applications ranging from teleconferencing to immersive multimedia. A key requirement for broadband array processing is a frequency-invariant spatial response, which ensures consistent directivity across wide bandwidths and prevents spectral coloration. Differential beamforming achieves such frequency-independent behavior by leveraging pressure differences between closely spaced elements in compact arrays. Traditional differential approaches, however, assume omnidirectional array elements, whereas real transducers often exhibit frequency-dependent directivity that can degrade performance if not properly modeled. To address this limitation, we propose a generalized modal-matching framework for frequency-invariant differential beamforming, applicable to arbitrary planar arrays of first-order directional elements. By representing the target beampattern as a truncated circular harmonic expansion and fitting it to the actual element responses, the proposed method accommodates arbitrary array geometries, element directivities, and orientations. The resulting framework enables the synthesis of high-order steerable beampatterns while avoiding restrictive layout assumptions. Simulations confirm that explicitly accounting for transducer directivity at the design stage yields accurate and robust performance across frequencies, array geometries, and noise conditions.