To answer your specific question, there is no way to be certain. The fraud should be picked up during the credit check process as your CC and DL will not be from the third country, but it might not and a clerical person at the pickup site might not notice or care.
But, what you may be assured of is that if you rent in violation of the contract and that standard document will always specify that it is void if incorrect information has been provided, that any insurance or other coverage associated with the contract will be void. That will be the subject of a check if there is a claim. If one uses one's own coverages, most also are void if the underlying contract is void.
So, all a risk calculation, some of which cannot be controlled or known before it may be needed.