database connection error for multisite WP in separate folder

I’ve been building a multisite WP network in its own folder on a server with an existing HTML website. The software was installed by Softaculus. The idea was that I just remove index.html and it cuts over to WP loveliness.

The WP sites work wonderfully if I access them as

website.com/wordpress

but if I access them as
website.com

then I get a Database Connection error.

The wp-config.php and index.php files are sym-linked into the root folder (to ensure they are the same) and .htaccess has +FollowSymLinks and all the WP rewrite rules are set as per what WP tells me:

Options +FollowSymLinks
Options -Indexes

RewriteEngine On
RewriteBase /wordpress/
RewriteRule ^index\.php$ - [L]

# add a trailing slash to /wp-admin
RewriteRule ^([_0-9a-zA-Z-]+/)?wp-admin$ $1wp-admin/ [R=301,L]

RewriteCond %{REQUEST_FILENAME} -f [OR]
RewriteCond %{REQUEST_FILENAME} -d
RewriteRule ^ - [L]
RewriteRule ^([_0-9a-zA-Z-]+/)?(wp-(content|admin|includes).*) $2 [L]
RewriteRule ^([_0-9a-zA-Z-]+/)?(.*\.php)$ $2 [L]
RewriteRule . index.php [L]

I’m assuming the database credentials are fine and the database is not corrupt since with a particular URL everything works brilliantly.

I’ve spent ages trawling the codex and other web sites but none of the advice is doing it for me and much seems inconsistent:

** I’ve tried adding the following to the wp-config.php

define('WP_HOME','http://website.com');
define('WP_SITEURL','http://website.com');

** Many instructions talk about changing the address in the Settings/General Site address (URL) but this doesn’t exist at network or site level.

I suspect some interaction between the URL rewrite rules and WP but I don’t understand it deeply enough to figure this out. Any help or tips appreciated.