In short: I’m trying to make a homing missle in 3D space which will fly in an arch towards the target and the closer it is to it, the tighter it’s arc will be (so it always hits the target). LookAt() makes them look very stiff instead of the desired arched motion.
Problem is - all the tutorials are for Rigidbodies which I am NOT using. Just simple Transforms withut any physics involved.
If you have some ideas how to achieve such a thing, I do really need your advice. 
Have you tried offsetting the rotation anchor by using LookAt with either an offset or by using it on a parent gameobject and then moving the child relative to it. You might find having the point of rotation near the front or back (as opposed to the center) really changes how it feels.
Failing that maybe writing some code that generates a spline between the two objects and then translating/ animating your projectile along that curve would allow you to better control it.
It would allow you to probably have a faster refinement/ iteration process as well as giving you more control into the behaviour. It would probably take a lot more effor though.