While radio measurements of extensive air showers have shown to achieve a high precision of $X_\mathrm{max}$ sensitivity, it has been shown that parameters beyond $X_\mathrm{max}$ can also be reconstructed. These shape parameters contain additional sensitivity to the hadronic physics in the shower as well as its mass composition. In this work, we showcase a reconstruction framework to recover the full longitudinal profile from realistic radio measurements. The framework is based on Information Field Theory that infers the full profile with a forward-based model, which uses a Gaisser-Hillas profile with weakly informative shower priors, SMIET with a template library to synthesise pulses at any event geometry, and a realistic antenna response and noise level emulating that of SKA-Low. We verify the self-consistency of our framework with $\sim 900$ events generated with SMIET with antennas placed on the $\vec{v} \times (\vec{v} \times \vec{B} )$ axis. The framework recovers the full profile within uncertainty and capture correlations between shower parameters. We yield an $X_\mathrm{max}$ resolution of $< 9$ g cm$^{-2}$ as well as resolutions of the width and asymmetry with minimal bias. The profile is also recovered with a bias of $< 4$% at all atmospheric depths $< 1200$ g cm$^{-2}$. We aim to apply this framework with pulses simulated from CoREAS with measured noise, ultimately extending the framework to realistic antenna layouts such as from LOFAR or SKA-Low.