Hans Christian Andersen died in 1875, the copyright is long expired but of course there might be other right-holders on the title or such.

Why would you want to write his story like this? The whole point of it is for the girl to die in cold, a critique of society's downlooking stance on poverty; just like Jonathan Swift wrote in his work a century earlier and looking at the number of children in poverty in Europe and the US I'd say there's no happy end in sight. 30% of the children in UK live in poverty, 21% in Germany, even Finland (which simply houses the homeless) has a rate of 10%.

> But there is an authoritative version of a particular writer's version of a particular story

Not necessarily. I heard this about, I think "Ulysses," but probably applies to most published books -- there are almost always changes between editions (if a books goes through multiple printings), differences in printings between different markets (even if those markets are in the same language), notes the author may have written at home but didn't get published, notes the author wrote on the review copy that got left out of the published version or got misunderstood or misspelled or otherwise improperly published...

A "text" turns out to be a lot less definitive of thing than it may at first appear.