How do we effectively proxy an HLS stream with limited connections, addressing lag and hanging issues with caching and latency tolerance?

How can we proxy an upstream HLS stream to the external network, if the provider restricts to only one connection at a time due to infrastructure concerns. I’ve managed to use proxy_pass for .ts, chunklist, and playlist files, and caching these media files effectively reduces the load on the upstream server.
When I attempted to cache the chunklist file for just one second, the stream sometimes experienced lag or hanging. If I avoid caching it, the upstream server may face heavy traffic. I must be overlooking something simple. Note that latency is not critical, and significant delays are acceptable. Do let me know about your suggestions on this.