Strange! When I switched from Alpine to [another Linux] that fixed my DNS errors with Node. So perhaps there is something else going on … maybe with the host machine itself 
Since Fly doesn’t charge for registry storage space, the size of the image is less of a concern (within reason) so maybe try a full Ubuntu/Debian as your base image. Shouldn’t be needed but worth a try while waiting for someone from Fly to debug. That would definitely rule out it being the base image.
I know Fly use Google’s DNS for resolving external domains and so maybe it’s related to that? Again, no idea why it would be. But that was from my fun with DNS a while back: