Was wondering about the thickness too, it does look like it doesn't have any and that will make a difference.
On the AO, lots of settings can be used to adjust the AO. A lower Max Distance will stop the rays looking so far for occluding surfaces, a higher Ray Directionality will keep the searching rays more focused rather than spread (and again "tighten" the results), and Colour Spread could help too. This video here goes through the settings, is using it as a dirt map but the principles of controlling the results of the AO are the same -
That said, AO might not be the solution here - generally it is used to highlight the shape of geometry by creating fake shadows to make things pop, but, a transparent / reflective material doesn't have any diffuse so doesn't catch shadows like that. Most every effect that lets us see the shape comes from reflection and refraction.
If you are free to make the scene any way you like, you could add a texture to the ground surface, and remove the back wall and let an HDRI image show there (like the object is placed on a kitchen countertop for example), could add some other objects on the counter top, even just more than one bottle. All of this would give things for the bottle to reflect and refract without having to fake anything. Attaching an image of the previous sphere (distorted it with an FFD modifier just so it has some shape), with the fake black cards and softbox turned off, and the Corona lights made less intense - with the background visible, their shape can be clearly made out without tricks!
You could use a very out of focus background for your HDRI to keep attention on the bottle.
(and yes, I was lazy and didn't fix the water in the rotated sphere - welcome to no gravity!)
PS - I did lower the reflection Glossiness to .5 on the plastic material here, so it has softer reflections than the water, felt it looked better with the background in full use.