Helena, 

here is a link to an answer from Céline previously. The short answer is ‘ because’ or ‘it’s colloquial, just the way it is ‘

Also a link to a Canadian site, which discusses this ( in French) with a useful reference to the dictionary Robert definition of ‘ un choix ‘

              Qu’est-ce qu’un choix ? Selon la définition du Robert, c’est la « décision par laquelle on donne la préférence à une chose, une possibilité en écartant les autres ». On évite alors d’écrire « j’ai deux choix ». 

The ‘ choice ‘ is not the option or options per se, but more akin the act or decision of choosing. 

Therefore, unlike in English  ‘the choice’ is considered singular and specific, countable and indivisible, as ‘one’ choice, choosing one option from one or several. The choice here is between ‘staying’ and ‘not staying’, no matter how many alternative ways of ‘not staying’, there may be. 

Ultimately, it is just a difference in the way French speakers view the nature of ‘ the choice ‘ compared to English speakers.

https://progress.lawlessfrench.com/questions/view/je-n-ai-pas-le-choix

 https://www.lapresse.ca/contexte/2023-03-19/en-un-mot/un-choix-difficile.php

What I do not understand is why it is LE in the first sentence but DE in the second. It is as though the Mais at the beginning of the sentence changes something. 

Helena - do you mean in Céline’s response ? I can’t see where that may be otherwise in the exercise you linked. If I am missing something, let me know and I will see if I can understand it and clarify.

I think the article from ‘lapresse’ indicates that it is not strictly correct grammar, hence the numerous options available and suggested to avoid using it.  On the other hand, every time you see the Académie or another ‘authority’ note that some expression or other shouldn’t be used, you can be sure it means that the expression is indeed in use, although not (yet) officially accepted as ‘ correct ‘ !

                    Pour éviter la locution « n’avoir d’autre choix que », calquée sur l’anglais to have no choice but, on peut préférer des expressions comme être astreint, contraint, tenu de faire quelque chose, forcé, obligé, être dans l’obligation de, devoir, ne pouvoir que, ne pouvoir faire autrement que. Le ministre a été obligé de répliquer. Elle ne pouvait qu’obéir. Je n’ai pas pu faire autrement que d’y aller. Il a dû faire face à l’opposition. Être forcé de partir, forcé à s’en aller. Je n’ai pas d’autre possibilité que partir.

I noticed that in French, one sometimes uses the specific where in English you would use the indefinite article. It's idiomatic to say c'est le bon choix ("it's a good idea"). I was looking for more such examples but, at the moment, they elude me.