Ignat, would you be able to provide some additional context? There are a number of possibilities to re-phrase your sentence, and without further clarification it may be difficult to decide how to express the author's intended meaning.
Is this a fiction or a non-fiction story? Is the emphasis on the sound, the noise made by an announcement which happens to be an announcement about registration for Flight Kabul-HK, or on the substance of the announcement itself? For example, 'the protagonist fell asleep and was woken up by a loud noise, which was an announcement...' or 'they announced that registration for flight Kabul-HK was open'; of course, it may be a combination of both. The author uses an inversion, and you propose the term 'rings out'. These suggest an emphasis on the sound itself. In either case, it might make sense to phrase it differently.
The author appears to insist that it is check-in/registration, while you seem to be inclined to change 'registration' to 'boarding'. When is the story taking place? In the past, on arrival at the airport, first, passengers would check-in/register for their flight; announcements of a kind 'check-in for flight N... is now open, go to counter N...' would have been frequent; at check-in, the airline would register those who are present to take the flight; the passengers would then go through security and wait in the boarding area until an announcement about boarding would invite them to board the plane; now, passengers check-in online; on arrival at the airport, passengers go through security and then - to the gate, where the airline would make announcements notifying the passengers that the flight is ready for boarding. If the author is adamant about 'registration', it may mean that it is material to the story, so, again, to decide whether you wish to refer to boarding or check-in, more context information would help. Of course, if it is a science fiction, or some imaginary period in time, then it is for the author to decide whether he/she wishes the announcement to be about check-in/registration or boarding (the protagonists would be doing different things in each case).
Also, the announcement, in the author's words, does NOT state that it is the start of 'registration'/check-in; the author simply tells us that (verbatim) there sounds an announcement about 'registration' for flight Kabul-HK. In the past, after the airline opened check-in/registration (or boarding) for a flight, announcements about check-in/registration (or boarding) would have been repeated continually. It may be material to the story whether the announcement refers to the start of..., or it is a repeated reminder about the check-in counter or the boarding gate being open.
In very general terms, in circumstances where stylistic and/or artistic devices are material to author's intended meaning, it is important to think through carefully how to convey the author's style in another language. What is technically possible in one language may not be possible in another, or not to the same extent (hence, the question about the use of inversion by the author; also, though the author probably speaks of 'an announcement', this will be clarified in the broader context).
Of course, if what you are working on is a simple non-fiction factual piece, there is no need for things to become so complicated.
P.S. For a very general introduction to translation work, see these recent articles about Elena Ferrante's translator Ann Goldstein in NYT and Esquire:
Every writer has challenges. Ferrante’s writing is very intense and
dense. ... Hers [sentences] are clear, but they’re going somewhere.
They're like, vroom! It's hard to capture that and still keep it
readable in English. Somehow Italian does that better, or it's easier
in Italian to do that. That's the biggest challenge.
Goldstein describes herself as a highly literal translator, an
approach that serves Ferrante’s idiosyncratic prose well, Reynolds
said. “It takes a great deal of humility and a great deal of courage
to represent so closely what an author wrote in the original