How techy are you and what level of handheld are you willing to get them? Retropie comes with a built in shutdown timer and you can set up cron jobs remotely to shut down after a time or even at certain times. Down side: You need to know how to set them up AND you are not going to get a machine that runs the latest and greatest.

You could go all out and get them a steam deck and set up crons but that's strictly local to the machine.