That version of Linux should have an NBD client (nbd.ko + nbd-client), so either swapping on NBD or using an external NBD device as a filesystem would be an option. He will need the corresponding oldstyle NBD server, eg: 'nbdkit --oldstyle memory 1G' to create a 1 gig RAM disk on the server side on port 10809.
(author) An interesting idea but the only `nbd*` I can find in the tree is `./usr/include/linux/nbd.h`; there's no kernel module or client pre-built. Also, I'm not sure this wouldn't exacerbate the issues with the Realtek just as badly as swap-over-NFS would, so the kernel upgrade would make more sense first.