What you might be missing is that the GPL doesn't allow such restrictions. The user is free to use, to modify and to redistribute modified or unmodified versions, provided that if they redistribute binaries, they offer the sources as well. AND, the GPL doesn't allow restrictions beyond that in the GPL itself. Thus, if someone (the contractor in this case) were able to get code under your negotiated GPL, they CANNOT be subject to limitations on redistribution and satisfy the GPL (whether they WANTED to redistribute further or not isn't the issue, besides, company officers and objectives can change, see what became known as SCOG), which would then invalidate the negotiated contract, which would make it an impossible contract from the beginning, because the conditions of it getting a somehow limited in ways the GPL strictly forbids GPL, are not possible to fulfill.
So the contractor cannot negotiate such a deal.
OTOH, if the third party in question (that would be providing the otherwise GPLed software) is sole copyright holder (or has otherwise procured relicensing rights from all contributors) on that software, then they could sell a proprietary license to it to the contractor -- not negotiating a GPL that's somehow not the GPL, but because they're sole copyright holder, negotiating terms different than the standard GPL entirely... for the right price... and indeed, this is or has been one of the funding models for ordinarily GPLed software -- see for instance MySQL and then Trolltech's GPLed Qt.
But that only works if that third party has full relicensing rights. Some projects (including the kernel) deliberately do NOT require such agreements, partly as a form of assurance to the community that they will always be freedomware -- guarding against the "Oraclizing" of MySQL, for instance. (Of course in that case, some of the people most loudly complaining about what Oracle is doing now, were the folks most loudly defending and insisting on the model when THEY were in charge -- before they sold out to... the Oracle they're now complaining about using the same tactics they used and defended!) For these projects, no third party could negotiate away the code freedom of the project.
That's my understanding as definite NON-lawyer but an interested community member, FWIW.