Remove Greenscreen Reflections? My work has a green screen that my boss purchased with no knowledge of green screens or how to use or what to do with them.

in Sept. 2014 he gave me the task to create employee introduction / interview videos to explain how some of our procedures work, and I used my knowledge of greenscreens and chroma-keying to do my best. I managed to create a couple of workable videos, where all of the green was easily edited out and the desired effect was achieved without much hassle. 

Recently my boss has asked me to create some product spotlight videos utilizing the green screen to place the item on a background of his choice. And here's where the trouble started. 



The Setup consists of:


2-flood lights
The screen and a stand for it


I have been using the lights to illuminate the screen and using the ambient light from the warehouse to light the object.

I have the object sitting on a black leather stool while it rotates (desired effect as requested by my boss) sitting about 2-3 feet away from the screen (this is the furthest away I can pull the object away before the screen edges are visible) 

The problems arise in editing, when I try to key out the green and add our background in. The black leather and the top of the product reflects so much that the keying eliminates it completely as the color technically matches that of the shadows in the screen its self, and BAM! The item is half transparent.

Modifying the key's parameters do nothing to help and my boss does not want to invest anything except my time and efforts into this.

After researching across the web, I have come to these conclusions:


Lighting is key, but we can not get more/better/more practical
Distance helps, but I can not move it much further because of screen size is too small to go further away and setting up in a warehouse
Reflective surfaces will be a pain anyways so make sure lighting is perfect and i can't get more lights. 


I have tried every idea I can think of to reduce reflection in photographs, but because it is not a static shot, there is much I am missing...

What else can I try?
You don't say what you're using to edit these videos, but there may be a better chromakey plugin for your platform than what comes natively. 

Also, again depending on your platform, most editors will allow you to crop the key area (garbage matte) so that the green screen doesn't have to cover the entire raster, just enough to back the product you're shooting, so you could put the screen as far away as you like.