The Shield was made for game streaming which would precisely enable you to connect the controllers to it and run the games on your PC. Nvidia has since stopped supporting that software but luckily there are alternatives. The Moonlight client and Sunshine host are direct implementations of what Nvidia used to provide or there's Steam Link if you're using Steam.
I know the DS4 controllers had similar connectivity issues on Windows which could be solved by changing power management settings. I'm not sure if this issue still applies to using DualSense on Linux.
Another method is to save the config as a template. The whole situation is so frustrating because before the switch to the Deck UI there was another tab which listed your configs for other games. I hope they bring that tab back at some point!