Ah, yes, the dreaded white object dillema. I would try to correct most of this with the camera and lighting. First, try bumping your exposure up to 1.3 or even higher if necessary. If that doesn't help, I'd play around with the lights. If getting a brighter bulb isn't an option, try moving them in closer, or angling them in different ways to try to really blast the background with light. Failing all that, you can sometimes get the rest of the way with Photoshop.I opened a few of your images up in Photoshop, and used the "levels" control to raise the whites as far as I could without nuking the whites on the toy. At that point there was still some grey in the background, so I used PS'es "quick select" tool to select just the background. A quick tutorial on quick select:
1. with the tool selected, hold the mouse button down and sort of paint around in the background until it's mostly selected. Because Jazz is white, parts of him will also get selected.
2. Hold down the alt key, and unselect the white parts of Jazz.
With a little practice, you can select just the background in about a minute or so.
Then, with just the background selected, open up the levels dialog and carefully amp up the whites until the background levels off at pure white. But really Photoshop should be a last resort. Good luck, and I look forward to your next attempt!