update: It seems I can only address the "possible to launch" part, but not the "possible to fuel first" or "possible to move over a big hole for the exhaust first" part.
Yes, with the operative word being "possible".
I have some first-hand experience, with the eye of a typhoon passing pretty-much overhead (that's what hurricanes are called here). The wind stopped, period. I could open my windows, it was quite calm. Unfortunately, even though it was night, I didn't see starts because I was too scared to leave my apartment, but certainly there is enough photographic evidence on the internet that the sky is clear. They fly planes in eyes.
Wikipedia's Eye (cyclone) confirms what we can guess, that the rotation and wind go to near zero in the center.
It also confirms that the eye gets wider as you go higher, better allowing for a curved trajectory that might not be the same direction that the system is moving.
There are plenty of risks, it's not guaranteed, but it is certainly possible.
Sources Maysak seen from the ISS 3. and File:Fly00449 - Flickr - NOAA Photo Library.jpg
Sources: Reddit Awesome view of the hurricane's eye from the ground (Photo Credit Jim Edds) and Eye of Hurricane Michael, just west of Mexico Beach, FL 10/10/2018