Should it be:
Bad weather doesn't exist.
The bad weather doesn't exist.
It depends on what you mean. Certainly bad weather does exist, so you could only be talking about an ideal place on the earth, if there is one, where it doesn't. So if you were talking about such a place (real or imagined), you could say
But if you were referring to some specific spate of bad weather that was expected but didn't arrive, you could say
But it would be more likely to say