You cuold try
--disk-cache-dir Use a specific disk cache location, rather than one derived from the UserDatadir. ↪
--disk-cache-size Forces the maximum disk space to be used by the disk cache, in bytes. ↪
and put the cache on a small ram disk. If that's all you have on the ram disk you can set it to discard everything on shutdown. In fact it may speed up your performance.
I used Data Ram Ramdisk not too long ago. It wasn't difficult to set up. Here's a list: http://en.wikipedia.org/wiki/List_of_RAM_disk_software
You can run FireFox Portable completely out of a ram disk. Chrome may be a bit more problematic.