We present an analysis of the NIRCam and MIRI morphological and structural properties of 80 massive ($ (M_ M_ odot )$\,=\,11.2\,pm \,0.1) dusty star-forming galaxies at $, identified as sub-millimetre galaxies (SMGs) by ALMA, which have been observed as part of the PRIMER project. To compare the structure of these massive, active galaxies to more typical, less actively star-forming galaxies, we defined two comparison samples. The first of 850 field galaxies matched in specific star formation rate and redshift and the second of 80 field galaxies matched in stellar mass . From the visual classification of the SMGs, we have identified 20\,pm \,5<!PCT!> as candidate late-stage major mergers, a further 40\,pm \,10<!PCT!> as potential minor mergers, and 40\,pm \,10<!PCT!> that have comparatively undisturbed disc-like morphologies, with no obvious massive neighbours on lesssim \,20\,--\,30\,kpc (projected) scales. These rates are comparable to those for the field samples and indicate that the majority of the sub-millimetre-detected galaxies are not late-stage major mergers, but have interaction rates similar to the general field population at $z$\,sim \,2--3. Through a multi-wavelength morphological analysis, using parametric and non-parametric techniques, we establish that SMGs have comparable near-infrared, mass-normalised sizes to the less active population, $ F444W $\,=\,2.7\,pm \,0.2\,kpc versus $ F444W $\,=\,3.1\,pm \,0.1\,kpc, but exhibit lower S\'ersic indices, consistent with bulge-less discs: F444W $\,=\,1.1\,pm \,0.1, compared to F444W $\,=\,1.9\,pm \,0.1 for the less active field galaxies and $n_ F444W $\,=\,2.8\,pm \,0.2 for the most massive field galaxies . The SMGs exhibit greater single-S\'ersic fit residuals and their morphologies are more structured at 2 relative to 4 when compared to the field galaxies. This appears to be caused by significant structured dust content in the SMGs and we find evidence for dust reddening as the origin of the morphological differences by identifying a strong correlation between the F200W$-$F444W pixel colour and the 870 surface brightness using high-resolution ALMA observations. We conclude that SMGs and both massive and less massive star-forming galaxies at the same epochs share a common disc-like structure, but the weaker bulge components (and potentially lower black hole masses) of the SMGs result in their gas discs being less stable. Consequently, the combination of high gas masses and instabilities triggered either secularly or by minor external perturbations results in higher levels of activity (and dust content) in SMGs compared to typical star-forming galaxies.