Yeah it sounds like Androids aggressive battery helper is causing this.
Is there any benefit to browsing private mode? Its only stopping history on your phone and not websites, so you can just set normal mode to not save history, cookies etc. Private mode is designed to be temporary in stopping your phone recording history and one button to close them all, so its not surprising that it doesn't save tabs, in fact its a feature
One option is to get an IPTV service and use it with an app where you can record the shows on disk (like TiVo I think you guys use, heard that a lot on older TV shows lol). Will only record future shows but tbh, a lot gets repeated on their other channels (https://www.channel4.com/tv-guide).
Using a VPN is another option but I recently couldnt load the app while using a lesser known VPN from within the UK so they seem hot on blocking them.
Another option is rent a cheap VPS from in the UK to act as your own VPN.
All of their shows is on their app, and they have a pretty cheap ad free option if you can figure that out too.