\section{Introduction and scope} Regularized immersed boundary methods couple Eulerian fluid variables to a separately represented structure through regularized interpolation and its adjoint force spreading \citep{peskin2002}. Parts~I and II of this series \citep{parti2026,partii2026} analyzed one conforming Cartesian realization under arbitrary changes of local enrichments, first for fixed interfaces and then for prescribed rigid motion. This paper makes the rigid motion an unknown. Translation and rotation are determined by Newton's equations, and the numerical body configuration therefore differs from the exact one. A convergence argument must control that difference jointly with the fluid error. The passage from prescribed to computed motion requires more than inserting a numerical trajectory into the estimates of Part~II. There the fictitious interior forcing was chosen to reproduce a known rigid acceleration. Here that acceleration is unknown. Surface constraints alone do not identify the interior fluid momentum with the displaced rigid momentum. Fictitious-domain formulations enforce interior rigidity and account for displaced-fluid inertia \citep[Section 4]{glowinski2001}; the interior-inertia issue is also explicit in regularized rigid-body IB formulations \citep[Section II]{kallemov2016}. Accordingly, we add regularized \emph{volume} moments to the regularized surface moments of Part~II. The discrete interior is only approximately rigid. Its defect is estimated rather than silently identified with zero. There is substantial prior analysis of fluid-driven rigid motion. San Mart\'{i}n et al.\ proved convergence of a fixed-mesh Lagrange--Galerkin method \citep{sanmartin2005}. Legendre and Takahashi derived joint fluid, rigid-velocity, and configuration error estimates in an ALE formulation \citep[Theorem 3.5 and Section 4]{legendre2008}. Discontinuous-density global formulations \citep{sanmartin2012} and free-body penalization convergence \citep{bost2010} provide other established approaches. In a fully Eulerian unfitted setting, von Wahl and Richter analyzed a coupled vector-valued parabolic equation and translational-motion ODE, including dynamic domain error \citep{vonwahlrichter2023}. Their model omits the fluid divergence constraint and body rotation, but its comparison of numerical and exact geometry is a direct methodological precedent. Thus neither coupled rigid-body convergence nor geometry comparison is claimed to originate here. Our energy argument builds on the usual cancellation of interface power in multiplier formulations \citep{boffi2015,boffigastaldi2017}. The background kinetic inner product uses positive \emph{excess} mass and inertia. These assumptions restrict the present analysis to sufficiently heavy bodies; they do not describe a physical instability of lighter bodies. An additional skew term is needed because the fluid equation uses the Eulerian Stokes derivative, not the Navier--Stokes material acceleration. We derive that term explicitly from the rigid interior. Its power vanishes, but omitting it would change the translational momentum equation. The numerical spaces and transfer principles retain the architecture of Parts~I--II. Dynamic-space Stokes projection estimates \citep{brenner2014}, stable adaptive transfers \citep{lundquist2021}, and moving-boundary remeshing analysis \citep{gawlik2015} supply the relevant context. We use exact orthogonal projection in a fixed fluid--body mass inner product. Every admissible local enrichment retains the entire coarse Cartesian streamfunction space. A constructed coarse comparison then removes any factor counting refinement and coarsening events. The method does not include an error-based marking rule, and the coarse grid itself is refined in a convergence sequence. Optimal adaptive fictitious-domain and regularized-forcing results \citep{berrone2019,heltai2023} address different questions. Regularized IB convergence results of Mori and Liu--Mori \citep{mori2008,liumori2012,liumori2014}, and prescribed-velocity Stokes estimates of Li et al.\ \citep{lipan2025}, motivate careful separation of bulk accuracy from boundary enforcement. Our constraints use exact continuum convolution and cell moments, rather than a particular sampled delta stencil. Regularization estimates \citep{heltai2020} and Sobolev trace and approximation results \citep{ern2021} are used explicitly in the defect bounds. The background velocity spaces are generated by conforming bicubic Hermite streamfunctions \citep{bfs2020}; the theorem is not a convergence theorem for a conventional staggered-grid AMR solver. Adaptive IB computations \citep{roma1999} and the dynamically adaptive DLM--IB framework of Zeng et al.\ \citep{zeng2022} predate this analysis and use different implementations. For moving fluid equations, evolving-space formulations \citep{alphonse2015,djurdjevac2023}, Eulerian unfitted analyses \citep{lehrenfeld2019,burman2022moving,vonwahl2022}, and ALE error estimates \citep{rao2025} provide further context. We use an auxiliary volume-preserving map to compare the exact and computed body configurations. This map is part of the proof, not a transport step in the algorithm. In particular, the square-root continuity of a trace with respect to interface displacement is not by itself a closed estimate for an unknown trajectory. Geometry alignment must first produce residuals linear in the configuration error. Computational studies also illustrate why boundary and force consistency deserve separate attention. He et al.\ developed an immersed pressure treatment to mitigate penetration through thin flexible structures in multiphase flow \citep{he2022multiphase}. Zheng et al.\ studied prescribed flapping near a free surface \citep{zheng2024swimmers}. IB calculations of backward-facing-step flow \citep{yang2021step}, airfoil transition \citep{chen2025airfoil}, and flexible canopies \citep{he2022canopy} give additional application context. These turbulent, multiphase, or deformable configurations are outside the present model; their numerical observations are not premises of our proofs. The result concerns one planar rigid body on an interval with positive wall clearance, positive excess mass and inertia, and a sufficiently regular exact Stokes--Newton solution. It is a numerical analysis conditional on that solution, not a new general existence theorem. The distinction between energy error after geometry alignment and energy error at the same Eulerian points is retained throughout: moving a gradient jump can weaken the latter estimate. Exact integration is assumed in the theorem. Numerical quadrature, finite-precision constraints, and selected algebraic identities are checked independently in the computational and Lean companions.