Many reasoning problems are based on the problem of satisfiability (SAT). While SAT itself becomes easy when restricting the structure of the formulas in a certain way, the situation is more opaque for more involved decision problems. We consider here the CardMinSat problem which asks, given a propositional formula $\phi$ and an atom $x$, whether $x$ is true in some cardinality-minimal model of $\phi$. This problem is easy for the Horn fragment, but, as we will show in this paper, remains $\Theta_2$-complete (and thus $\mathrm{NP}$-hard) for the Krom fragment (which is given by formulas in CNF where clauses have at most two literals). We will make use of this fact to study the complexity of reasoning tasks in belief revision and logic-based abduction and show that, while in some cases the restriction to Krom formulas leads to a decrease of complexity, in others it does not. We thus also consider the CardMinSat problem with respect to additional restrictions to Krom formulas towards a better understanding of the tractability frontier of such problems.