but it's a single line leading to a black-box C interface that takes so many days

And there in lies the problem -- without cooperation its not going to work well

Most computers have a "hibernate" mode, aka syspend-to-disk, and it works when every program is suspended including the operating system, and the memory is dumped to a pagefile on the disk

hibernating a single process can get very tricky ... but I imagine if you've got the RAM, and you run a virtual machine, you can simply hibernate the virtual machine ...

Good luck