Multipath wireless access aims to seamlessly aggregate multiple access networks to increase data rates and decrease latency. It is currently being standardized through the ATSSS architectural framework as part of the fifth-generation (5G) cellular networks. However, facilitating efficient multi-access communication in next-generation wireless networks poses several challenges due to the complex interplay between congestion control (CC) and packet scheduling. Given that enhanced ATSSS steering functions for traffic splitting advocate the utilization of multi-access tunnels using congestion-controlled multipath network protocols between user equipment and a proxy, addressing the issue of nested CC becomes imperative. In this paper, we evaluate the impact of such nested congestion control loops on throughput over multi-access tunnels using the recently introduced Multipath DCCP (MP-DCCP) tunneling framework. We evaluate different combinations of endpoint and tunnel CC algorithms, including BBR, BBRv2, CUBIC, and NewReno. Using the Cheapest Path First scheduler, we quantify and analyze the impact of the following on the performance of tunnel-based multipath: (1) the location of the multi-access proxy relative to the user; (2) the bottleneck buffer size, and (3) the choice of the congestion control algorithms. Furthermore, our findings demonstrate the superior performance of BBRv2 as a tunnel CC algorithm.