I believe you need to use tweak the config/general.php file in order to have a baseUrl for a site that is a different domain name.
'siteUrl' => [
'english' => 'https://www.mysite.com/',
'japanese' => 'https://www. myjapanesesite.com/',
See documentation at: https://docs.craftcms.com/api/v3/craft-config-generalconfig.html#siteurl
As far as I know, using the @web alias is discouraged if its determined automatically (which is the default behavior).
If you override it with a hardcoded value in production I think you should be fine.
After checking the documentation, the warning is now pretty dissuasive against using the @web alias :
When you actually use it in a site's base URL ...