In this paper, we consider a team of mobile robots executing simultaneously multiple behaviors by different subgroups, while maintaining global and subgroup line-of-sight (LOS) network connectivity that minimally constrains the original multi-robot behaviors. The LOS connectivity between pairwise robots is preserved when two robots stay within the limited communication range and their LOS remains occlusion-free from static obstacles while moving. By using control barrier functions (CBF) and minimum volume enclosing ellipsoids (MVEE), we first introduce the LOS connectivity barrier certificate (LOS-CBC) to characterize the state-dependent admissible control space for pairwise robots, from which their resulting motion will keep the two robots LOS connected over time. We then propose the Minimum Line-of-Sight Connectivity Constraint Spanning Tree (MLCCST) as a step-wise bilevel optimization framework to jointly optimize (a) the minimum set of LOS edges to actively maintain, and (b) the control revision with respect to a nominal multi-robot controller due to LOS connectivity maintenance. As proved in the theoretical analysis, this allows the robots to improvise the optimal composition of LOS-CBC control constraints that are least constraining around the nominal controllers, and at the same time enforce the global and subgroup LOS connectivity through the resulting preserved set of pairwise LOS edges. The framework thus leads to robots staying as close to their nominal behaviors, while exhibiting dynamically changing LOS-connected network topology that provides the greatest flexibility for the existing multi-robot tasks in real time. We demonstrate the effectiveness of our approach through simulations with up to 64 robots.