Bundling multiple access technologies increases capacity, resiliency and robustness of network connections. Multi-access is currently being standardized in the ATSSS framework in 3GPP, supporting different access bundling strategies. Within ATSSS, a multipath scheduler needs to decide which path to use for each user packet based on path characteristics. The Cheapest Path First (CPF) scheduler aims to utilize the cheapest path (e.g. WiFi) before sending packets over other paths (e.g. cellular). In this paper, we demonstrate that using CPF with an MP-DCCP tunnel may lead to sub-optimal performance. This is due to adverse interactions between the scheduler and end-to-end and tunnel congestion control. Hence, we design the Adaptive Cheapest Path First (ACPF) scheduler that limits queue buildup in the primary bottleneck and moves traffic to the secondary path earlier. We implement ACPF over both TCP and DCCP congestion controlled tunnels. Our evaluation shows that ACPF improves the average throughput over CPF between 24% to 86%.