  When collecting spectroscopic data on at least four isotopes, nonlinearities
in the King plot are a possible sign of Physics beyond the Standard Model. In
this work, an improved approach to the search for hypothetical new interactions
with isotope shift spectroscopy of few-electron ions is presented. Very careful
account is taken of the small nuclear corrections to the energy levels and the
gyromagnetic factors, which cause deviations from King linearity within the
Standard Model and are hence a possible source of confounds. In this new
approach, the experimental King nonlinearity is not compared to the vanishing
prediction of the Standard Model at the leading order, but to the calculated
full Standard Model contribution to King nonlinearity. This makes searching for
beyond-the-Standard-Model physics with King linearity analysis possible in a
very-high-precision experimental regime, avoiding confounds. The bounds which
can be set on beyond-the-Standard-Model parameters remain limited by the
uncertainties on the small Standard Model nuclear corrections which cause King
nonlinearity. Direct comparison between theory and experiment on a single pair
of isotopes is advocated as a more suitable approach for few-electron ions.
