The solution I have found to this is:

1. To invent a file extension for the file, so traffic-advice is renamed to traffic-advice.ta-json (an invented file extension ... traffic advice JSON)

2. Use .htaccess to rewrite requests for traffic-advice to traffic-advice.ta-json

3. And again in .htaccess, use AddType to set the required MIME type for ta-json files

.htaccess therefore includes:

<IfModule mod_mime.c>
AddType application/trafficadvice+json ta-json
</IfModule>

<IfModule mod_rewrite.c>
RewriteEngine On

RewriteRule ^\.well-known/traffic-advice$ .well-known/traffic-advice.ta-json [PT]

# continues with other RewriteRules
</IfModule>

The traffic-response file lives in .well-known, only it's renamed with my 'invented' file extension to act as the trigger for setting the MIME type.

In Chrome and Firefox I can now see in the response header:

content-type: application/trafficadvice+json