They are a hard fighting fish and large fish at that. They have a lot of blood compared to other fish I've cleaned which means they need a lot of air. Holding them out of water for any amount of time after you reel them in hurts them. An air actor in a live we'll probably is enough. They need to continue swimming and getting water thru their gills to live.
Most of the stripers I've caught have given it all they had before I got them in the boat. I haven't had much luck with catch and release once the water gets warm. I've had a few Texoma trips cut short once I got two overs and everything was coming up over 20 inches. Not very sporting to keep fishing with floaters around your boat. Maybe if you horse them in on bigger tackle it's different.
Stripers are native to cooler saltwater. In general saltwater fish need highly oxygenated water to survive. Your livewell doesn't even come close to providing what they need.