Thank you for waiting, while I performed some further tests and research.
I investigated your website's clone on my test server and noticed that even with all extra plugins disabled and with the default WordPress theme, the WordPress export feature fails, as you noticed. The server I used for testing had exceptionally high memory and processing resources allocated, but it kept giving the 'out of memory' errors.
The database of your website is huge, especially the posts and the posts meta table and the titles of the posts consist of long and non-standard characters. This all adds up to the complexity of the database queries.
Based on this, here are a few of my recommendations:
Option 1:
On your website, I've created a page 'Lieux accessibles - Relationship data', that uses a view named 'View to show Lieux accessibles - Relationship data'.
This view shows the tabular list of existing 'Lieux accessibles' posts along with the title of the related 'Quartier' post if the relationship exists.
For each record, it also shows a relationship form 'Lieux et Quartiers' that can be used to connect the current 'Lieux accessibles' post to an existing 'Quartier' post.
To avoid memory and performance issues, I've set this view to show records with '50' items per page.
I can understand this is not ideal, as you'll still have to add the relationships one by one, but I'm sharing this as one of the possible approaches since it doesn't require any custom code or paid third-party plugin add-ons.
Option 2:
You'll find another page named 'Lieux accessibles - export CSV data' that uses a view 'View to show Lieux accessibles - export CSV data'.
This view generates a comma-separated list of existing 'Lieux accessibles' posts, with their IDs, titles, and the titles of the related 'Quartier' post if the relationship exists.
To avoid memory and performance issues, I've set this view to show records with a '50' items per page.
You can copy/paste this data manually into your CSV file, which will solve half of the challenge, which is the export of the data.
Note: in your other support ticket you mentioned, that you tried the 'WP All Import' plugin. But did you try its paid version? The custom field import is only supported by the paid extension.
Once, your export file is ready with the '_toolset_associations_lieu-et-quartier' column for the custom field values for the related 'Quartier' post titles and if you face any challenge in mapping this column with the custom field during the import, you're welcome to let me know through a new ticket.
( just mention this ticket's reference and we'll have the background information )
Option 3:
Another alternative is hiring a professional from our list of recommended contractors, who is proficient with PHP, WordPress, and Toolset.
( ref: https://toolset.com/contractors/ )
A professional can create custom functions, which can be used to export the CSV file with the desired data and then also for the import process, ensuring that these operations are performed in small batches to avoid server/memory issues.