"Please don't go to that mountain in the winter."
I'm a native speaker. To me 冬に and 冬には would be a little strange because then に is being used twice to point to the same verb, making the intent ambiguous.
That being said, both 冬はその山に or その山は冬に are acceptable; as per usual, the main difference is connotation (is the topic the mountain or the winter?).
I feel that it could be では to say during the winter, but I'm not sure. I'm still learning grammar :/