IT's not rat, you can tell by the fact the "eye sockets", meaning the two holes in the centre of the top/bottom surface of the skull are much too small for the size of the skull and they are too close together. Also, if that was a rat skull, the rat would be at least 10" long, not including tail.
The sort of flowing ares near what many people are calling the "nose" indicates some sort of adaption to accomodate tubes or something like that. Perhaps, wind pipes or a propulsion sytem like octopi and cuttlefish.
No teeth left, no way it's a gator 'cause there aren't any around here..
It's not fake