The nanophotonic signaling technology enables efficient global communication and low-diameter networks such as crossbars that are often optically arbitrated. However, existing optical arbitration schemes incur costly overheads (e.g., waveguides, laser power, etc.) to avoid starvation caused by their inherent fixed priority, which limits their applicability in power-bounded future many-core processors. On the other hand, quality-of-service (QoS) support in the on-chip network is becoming necessary due to an increase in the number of components in the network. Most prior work on QoS in on-chip networks has focused on conventional multi-hop electrical networks, where the efficiency of QoS is hindered by the limited capabilities of electrical global communication. In this work, we exploit the benefits of nanophotonics to build a lightweight optical arbitration scheme, FeatherWeight, with QoS support. Leveraging the efficient global communication, we devise a feedback-controlled, adaptive source throttling scheme to asymptotically approach weighted max-min fairness among all the nodes on the chip. By re-using existing datapath components to exchange minimal global information, FeatherWeight provides freedom from starvation while resulting in negligible (< 1%) throughput loss compared to the best-effort baseline optical arbitration. In addition, FeatherWeight provides strong fairness, performance isolation, and differentiated service for a wide range of traffic patterns. Compared to state-of-art optical arbitration schemes, FeatherWeight reduces power consumption by up to 87% while reducing execution time by 7.5%, on average, across SPLASH-2 and MineBench traces, and improving throughput on synthetic traffic patterns by up to 17%.