Communication solutions for passengers on public transport vehicles, in the form of on-board networks connected to the Internet via a mobile router (MR) and a wireless link, are increasingly offered by public transport providers in many countries. An important challenge in such networks is to guarantee a fair access to the scarce wireless bandwidth by all on-board users, particularly in networks with multiple wireless interfaces. Accordingly, we consider an active fairness control scheme, where the MR transparently overrides the TCP advertised receiver window size to limit the transmission rates of on-board TCP connections to their fair share. We focus on the overhead that this scheme places on the mobile router and the wireless interface, due to the need for the MR to estimate the round-trip time (RTT) of each on-board connection. We investigate the performance of three strategies that differ in the timing of RTT estimations, and show that a fairness target-based strategy, which continuously monitors the ongoing fairness index and invokes an RTT reestimation whenever it drops below a predetermined value, consistently achieves a significant improvement in fairness for only a mild overhead cost.