How to lock rotation in a specific way?

I'm trying to make a simple top down game, and at the moment the player moves with aswd, etc, and I want the player to always face the mouse. for example, I have [^] <-- sprite The carrot should face the mouse always. My script so far is thus:

 void TurnBody()
     {
         Vector3 mousePos = new Vector3(Input.mousePosition.x, Input.mousePosition.y, 0);
         transform.LookAt(Camera.main.ScreenToWorldPoint(mousePos), Vector3.forward);
         //transform.rotation = Quaternion.Euler(0, 0, -(transform.rotation.eulerAngles.z));
         //rb2d.constraints = RigidbodyConstraints2D.FreezeRotation;
         //transform.eulerAngles=new Vector3(0,0,transform.eulerAngles.z);
     }

I've tried using atan2, various quaternion stuff, which did not work. The uncommented works great, except for some reason the sprite is not "flat" unless the mouse is very close to the sprite. The commented is some stuff I tried in order to lock the rotation on the various axes, but when I do, the rotation is no longer smooth and becomes jerky, and often goes in the wrong direction. (opposite to the mouse) I'll keep trying things, though any constructive input would be appreciated, thanks. I am also open to entire other solutions to the problem.