The Access Traffic Steering, Switching, and Splitting (ATSSS) technology, currently under standardization by the 3rd Generation Partnership Project (3GPP), is designed to enhance wireless access through the bundling of both 3GPP and non-3GPP access. One implementation option involves tunneling data through the Multipath QUIC (MP-QUIC) protocol. The decision of which access network each packet is transmitted on can greatly impact the performance of a network flow, and may also impact the performance of other flows. While Active Queue Management (AQM) in the network could normally manage such interactions, such techniques are limited in a shared multipath-tunnel context, since all traffic in the tunnel share a single connection. This paper introduces a multi-flow, multipath tunneling framework using MP-QUIC. It employs four distinct scheduling policies for packet scheduling decisions. We investigate how such scheduling policies interact with each other, and how they may be combined in order to achieve trade-offs in terms of latency, interactivity, and throughput. Through an extensive investigation in both emulated and real-world environments, we show that Quality of Service (QoS) can be significantly improved by smart combinations of scheduling policies.