Brad when oil stains a synthetic fiber that badly (my guess is its olefin, but I have seen it on other fibers), its stained the fiber just like food dyes stain nylon or wool...or worse. You can only hope to "lessen the intensity" of the stain (a nice weasel word that says you won't get most of it out) with either oxidizing agents (think Boost All by Bridgepoint or any of percarbonate boosters) or give it a bit of an "encap cover" by filling in the damaged areas with the "good residue" of an encap product. There seems to be some removal as well, depending on the formulation of the encap, but I'm not yet satisified with the answers I've read about that to say how it works, I can just say that it seems to work.
OK, I'm being verbose... imagine that. Here are the "Cliff Notes":
1. Its abuse of the carpet. Not your fault, few if any can do better that you appear to have done with the equipment and cleaning products you've used. Some solvent boosters get a bit more out, but I've not seen spectacular results with anything I've tested.
2. If you feel the need to try more, add an oxidizing booster that contains Sodium Percarbonate.
3. You can also "encap" afterward. There is good evidence that this will help.
4. Don't promise that steps #2 and # 3 will do much more than improve it a little ...remember "lessen the intensity".
5. If they want more, remind them of #1.