The discrete choice paradigm of random regret minimization (RRM) has been recently proposed in several choice contexts. In the route choice context, the paradigm has been used to model the choice among three routes, to define regret-based equilibrium in risky conditions, and to formulate regret-based stochastic user equilibrium. However, in the same context the RRM literature has not confronted three major challenges: (i) accounting for similarity across alternative routes, (ii) analyzing choice set composition effects on choice probabilities, and (iii) comparing the RRM model with advanced RUM counterparts. This paper looks into RRM-based route choice models from these three perspectives by (i) proposing utility-based and regret-based correction terms to account for similarity across alternatives, (ii) analyzing the variation of choice set probabilities with the choice set composition, and (iii) comparing RRM-based route choice models with C-Logit, Path Size Logit and Paired Combinatorial Logit. Results illustrate the definition of RRM-based route choice models with correction terms within the regret function, show their lack of robustness with respect to the choice set composition, and present their positive performance when compared to advanced RUM-based models.