Hi first please do not let the mouse go outside. Even though "it is just a mouse" it is a domesticated mouse not wild. You can get a small critter keeper to put the mouse in. If you plan on attempting to feed it to your snake again you can offer it food from your kitchen...fresh veggies...my rat loves carrots, dry cereal,dry fruit, low protein cat food, and of course fresh water.
As to the mouse stressing out your snake...
How long have you had the snake?
You should wait a good week with no handling before attempting to feed. Your snake is not going to starve in just a couple weeks and repeated attempts at feeding will just stress out your snake more.
I remember reading your previous post about your snake killing the prey and then not eating. I feed live...I watch to make sure that Zeke has killed the rat and then I walk away. Standing over him makes him back away from the kill.
If after 10-15 minutes Zeke has not killed the rat it comes out and I wait a week to try again.
There are times that Zeke has gone into his hide and "hid" from the snake. I have had him long enough to know if he is in there to "ambush" his prey or because he is not interested.
I know your frustration...Zeke went off feed this winter for almost 2 months and then spent the next 4 mths eating sporadically. He is currently back to slamming his rats with gusto every week
I guess what I am saying is just leave your little one alone as much as possible and try again next week....believe me..it is more stress on you than on the snake.