GFCI equipped machines into GFCI receptacle
There is no reason one GFCI should cancel out another GFCI. Don't trust that the device is GFCI-protected unless it specifically states so. As Ed Beal mentioned in his comment, it seems likely that the device is actually leaking current, and thus causing a (not unreasonable) trip in your GFCI-breaker. 

My advice would be to call the maker of the device to ask specifically why it states not to plug into a GFCI outlet. They should also be able to tell you what they recommend you use, since you don't have a non-GFCI outlet.

If, as you mention, the GFCI protection is in your breaker, you are correct that there aren't any outlet tricks you could perform to create a non-GFCI outlet, and in most cases you should avoid removing this protection anyways.

UPDATE: I'm updating my answer in response to more information from the OP in comments.

Since you have confirmed the device has it's own GFCI, I would recommend calling your local planning/permitting authority, or a licensed electrician, and asking them what your options are. It may be possible to install a dedicated outlet, or perhaps hard-wire this device, within code. Without knowing exactly what your local code is, I can't recommend any other action.