What level of rigor is needed to determine relative importance? I'm performing dominance analysis (using domir) not to predict/forecast but only to report on the relative importance of a set of independent variables. Do I still need to analyze with split datasets (train and test) and make sure that relative importance is stable over multiple random splits?

Dominance analysis is, in my view, an extension of the underlying model that adds information about predictive utility.  If the underlying model is stable as determined by whichever means the analyst finds defensible, then I don't see a good reason to attempt to assess the stability of the dominance analysis based on that model.
I would then recommend assessing stability on, for example, the lm()/linear regression model results directly.  If they appear to be sufficiently stable and seems reasonable to proceed with, then a dominance analysis using that linear model is justified.
Again, a dominance analysis provides more information about predictive utility of predictors related to a pre-selected model.  I see applying training-testing/cross-validation methods as a model selection step and better applied to a model and not a dominance analysis on which it is based.
