Past lattice simulations tentatively suggested that the spectrum of observable particles in BSM theories is qualitatively different than perturbatively expected. The discrepancy can be traced back to nontrivial field-theoretical effects arising from the requirement of gauge invariance. A perturbative but gauge-invariant approach working directly with bound states (as proposed by Fröhlich, Morchio and Strocchi) could provide a solution, by allowing composite-state effects to be treated within a perturbative framework. We consider a toy GUT-like theory – SU(3) Yang–Mills coupled to a scalar ‘Higgs’ in the fundamental representation – and expand on previous work by providing the most comprehensive spectroscopy to date, including all channels up to spin 2. Our results strongly support the general conclusion that the elementary spectrum is not an adequate proxy for the low-energy spectrum of a GUT, and also suggest a possible analytical approach to BSM model-building via the Fröhlich–Morchio–Strocchi approach.