I just paralyzed a baby mouse while trying to capture it with a bowl
Why did you paralyze a baby mouse ?
I was thinking that it wouldn't hurt it.
Well that sounds like an honest mistake.  At this point, there's probably nothing you can do for the mouse and it'd be better for it if you just killed it.  That will spare it further suffering.
Well, I guess that's on me. I mean it's not like it was my fault, right? I just wanted to keep it in a cage.
It was not your fault but I hope you learned not to capture mice like this. You better euthanize that mouse now, that'd end that poor soul's misery.