The third-party company claims that the Linux Driver is derivative
work of the driver they previously developed for OSX, so they say GPL
does not apply here.
If they link to the Linux-kernel, they link against GPL-software, so they are obliged to respect the terms of the GPL. It doesn't matter if the software was previously developed for OSX. In the worst case the licenses conflict and the Linux version is therefore illegal to distribute. Generally, all licenses of products combined here are to respect. If they aren't compatible, you cannot distribute this derivate.
But, if the driver doesn't directly link to the kernel, that may not apply. I remember in the past some graphics drivers (I think NVidia, but aren't sure) had two parts. One GPL-part included in the kernel and offering an interface for the second part, that was proprietary. I don't know how legally stable this construct is, but it may work.
As I don't know in which way the driver you talk about works, I cannot say which applies here.
But even if case 1 applies, you cannot release the source code of others without their consent. If case 1 applies, you simply cannot distribute this solution.
As a side-note: the GPL always allows you to use such software internally without releasing the source, but as you talk about customers it isn't limited to internal use.