The NEC is not a design manual.
If the NEC says it's 'OK' to use a 15 amp breaker in place of a 1 amp fuse, that just means is legal in jurisdictions that have adopted the NEC as their electrical rule book.
That doesn't make it a good practice or a good design.
One of my favorite code instructors had a saying that the electrician that followed the NEC to the letter was basically the worst electrician allowed by law. I think the saying is funny and to a certain extent, very true. Others, I am sure, will disagree.
Let's look again at the OP's question:
If you have a 1 amp fuse protecting a device like a transformer. And you want to replace it with a miniature breaker. Do you size the breaker at 1 amp also and if so how do you determine the trip curve?
The OP didn't ask what the legal minimum was.
I would use the same amp breaker as the replaced fuse and try to get the trip curve from the manufacturer if possible. If a trip curve is a requirement, then I would look for a breaker that I could get that had that info.
Even before that, I would want to know why the client wants to do away with the fuse before I modified it for a breaker. There may be no good reason to do so, after a bit of discussion. If the reason is excessive tripping, then there is more to look at than putting in a breaker.