I notice you're referring to "benzene" rather than "benzine." Here in the US, I don't know of any common sources for benzene, which is nasty stuff, but benzine with an "i" is different, basically some variant of white gasoline, lighter fluid, Coleman fuel, etc., and pretty benign. It's what fire eaters use. Things might be different esewhere, but I think if I were looking in the US for a relatively non-hazardous petroleum solvent in quantities greater than a can of Ronsonol, I'd check out camping supply stores for stove fuel. Toluene is a benzene with the "e" derivative, and seems to have many of the same risks.
PS. It occurs to me to wonder whether "water white" kerosene might be an alternative too. Some sewing machines (old Elnas come to mind) specify kerosene as a lubricant for the rotating shuttles, where even the thinnest oil is too oily. It certainly will mix with oil and thin it, though I don't know if it would be thin enough. Here in the US, it can be hard to find kerosene that has not been adulterated, either with aromatics for lamps, or with dye to indicate the non-payment of highway taxes, but it can be found.
