Put a white sheet between you and the trophy - some distance from the trophy, but basically "all around". Cut a rectangular hole in it that is about the size of the trophy. Use a long lens, and shoot the trophy through the hole. Now most of what is reflected will be "white sheet", with just a small hole in the middle where you were standing. If you further put a second white sheet where you are standing, and just point your lens through it, you will reduce the reflections to a small dot in the middle of the trophy. The rest can be done with photoshop.

Reason for two sheets is this: you really want 360 degree coverage, but you don't have a "white room". So you make a "small white room", and look through the keyhole (as seen from the trophy) at a slightly larger "sheet". Here is an approximate diagram:

You probably want to make sure you illuminate your "second sheet" to make it appear roughly equally bright as the inside of the first sheet. Play around with the positioning of your flash guns to get the desired effect.

update see http://blog.calumetphoto.com/2013/04/photographing-metal-reflective-surfaces/ for some more info / examples

response to icycle regarding "using a long lens far away doesn't work" - the following diagram shows that this is true for a flat mirror (the point on the surface where the reflection of the edge appears is the same regardless of distance), but it is NOT true for a curved object: a lens of the same size will seem smaller relative to the object when it is further away. One way to think about it: you always see "the entire room" in miniature in the reflection in the object (I use a sphere for simplicity). When you are very close, your lens looks like a large part of the room; when you are very far away, it becomes much smaller.

If you need to use a larger lens because you are further away, this is no longer true - in fact if you are trying to shoot with a constant F number lens, since F number is ratio of diameter to focal length, and magnification scales with focal length, then the lens needs to get bigger as it gets further away by just the right amount to keep its reflection constant…

One more thing you can do with the given setup: if you don't need to shoot with lens wide open, consider a piece of card that is white on the front and black on the back, and cut a circular hole in it. Put it right in front of your lens. Depending on the lens construction, you may be able to get a good picture through this hole while the "lens blob" in the reflection will look smaller. Experiment with the f number you use - in principle you will be restricting the lens light gathering capacity which means that you can't trust the f number, and you also have to worry about DOF and how the edge of the circle may become visible in the picture (depending on the view angle of the lens, and the distance from the card to the optical center). You need to paint the inside edge of the hole matte black to prevent haze.

PS - great job on the picture - even with the "larger" lens reflection, it came out pretty clean looking! Happy to know the customer liked it too.