If it would be August then I guess you could make a deal with one of the cheap cars from the Mongol Rally (which actually finishes just North of Mongolia)
I have read Mongolia also has some specific rules officially, about if one person is entering with a vehicle and another person is leaving. But it should hopefully not be a problem if that first person entering Mongolia does not say anything about it, and just gives the impression that he/she will be leaving with the car again.
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