How do I increase rate of FixedUpdate to ~1000x per second?

I made a drawing functionality that draws dots onto a Texture2D using SetPixel.
It uses FixedUpdate to draw the dots:

void FixedUpdate()
{
    //Return if canvas menu is not open
    if (canvas.enabled == false)
    {
        return;
    }

    //Check if left mouse button is down
    if (Input.GetMouseButton(0))
    {
        //Get position of mouse
        Vector3 mousePosVector3 = Input.mousePosition;
        Vector2 mousePosVector2 = new Vector2(mousePosVector3.x, mousePosVector3.y);

        //Check if mouse is in canvas
        if (mouseInCanvas(mousePosVector2))
        {
            paintAtMouse(mousePosVector2);
        }
    }
    else
    {
        lastMousePos = new Vector2(0, 0);
    }
}

However, when the mouse moves too fast, gaps between the dots appear like so:
image

I have tried using linear interpolation to fill the gaps, but it causes the line drawn to be very angular unlike the curved motion of the mouse that drew the line.
I have also set the Fixed Timestep to the lowest value it allows:
image

I would appreciate any suggestions on how to tackle this problem!