In this article, we consider a class of strictly hyperbolic triangular systems involving a transport equation. Such systems are known to create measure solutions for the initial value problem. Adding a stronger transversality assumption on the fields, we are able to obtain solutions in $L^\infty$ under optimal fractional $BV$ regularity of the initial data.Our results show that the critical fractional regularity is $s=1/3$. We also construct an initial data that is not in $BV^{1/3}$ but for which a blow-up in $L^\infty $ occurs, proving the optimality of our results.