I haven't done an oil print (yet) but I do now a little bit about paint that might help.
The difference between the different oil based inks/paints is usually the binder they are mixed with. I've heard that you can use oil paint, but it is a very different constancy from a litho or intaglio ink. You could try adding sun thickened or polymerized linseed oil to artists grade oil paint if you absolutely can not find a litho or oil based etching ink. Also litho inks often have resins and waxes to further their water resistant properties.
Also litho inks (tend to) use pigment which will not be soluble in water. Oil paint isn't expected to get wet, so research into specify pigments is needed. Another thing to avoid is 'water soluble oils' sounds obvious, but they would desolve in water.
In printmaking classes we would often take a xerox copy, saturate it with gum arabic, then ink it up with oil based intaglio inks just like an oil print is done. So I wouldn't worry too much about using an intaglio ink in place of litho.
Anyways, that's my 2 cents, hope it was helpful.