Jet contrails reflect solar energy away from the Earth during the day. They also reflect heat back to Earth at Night. If polar flights were banned during the northern winter (polar night), would there be a net cooling effect on the Arctic?
Contrails also reflect heat back to the earth during the day. The net effect is likely warming for most circumstances. However, it is a small effect. The effect of contrails above the polar region seems to be almost none - but if that is because the ground is already highly reflective, because contrails are hard to see there or because not that many planes fly there, I couldn’t say (On the Trail of Contrails | Earthdata).
So, what will happen due to a ban? If the flights still happen, but taking a longer route, warming due to more carbon being burned.
If the flights don’t happen, less warming because less carbon is burned.
The contrails don’t really factor into this.