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Tendon-driven aerial continuum manipulators (TD-ACMs) combine the maneuverability of uncrewed aerial vehicles (UAVs) with the compliance of lightweight continuum robots (CRs). Existing coupled dynamic modeling approaches for TD-ACMs incur high computational costs and do not explicitly account for aerial platform underactuation. To address these limitations, this paper presents a generalized dynamic formulation of a coupled TD-ACM with an underactuated base. The proposed approach integrates a strain-parameterized Cosserat rod model with a rigid-body model of the UAV into a unified Lagrangian ordinary differential equation (ODE) framework on $\mathrm{SE}(3)$, thereby eliminating computationally intensive symbolic derivations. Building upon the developed model, a robust dual-camera image-based visual servoing (IBVS) scheme is introduced. The proposed controller mitigates the field-of-view (FoV) limitations of conventional IBVS, compensates for attitude-induced image motion caused by UAV lateral dynamics, and incorporates a low-level adaptive controller to address modeling uncertainties with formal stability guarantees. Extensive simulations and experimental validation on a compact custom-built prototype demonstrate the effectiveness and robustness of the proposed framework in real-world scenarios.