Problems with Multisite WordPress Installation

I am encountering issues while setting up a WordPress Multisite Network on my company’s production server, which is globally accessible using the subdomain localization.exe.sk (note: we do not own exe.sk; it hosts a different site).

It all started when I tried to move new wordpress site (a multisite network, each site is a language variation of the same site), from testing server to a production server.
I did it by exporting wordpress database from testing server and importing it to production server (alongside with the old database from the old web), and then copying new site’s wordpress files from testing server to production server (while backing up old web files somewhere else), also ensuring all privileges to files, and in database, that db creadentials are correct, to ensure that everything is set up correctly.
Also, in wp-config file, i changed the DOMAIN_CURRENT_SITE to localization.exe.sk.

After transferring the Multisite network to the production server as stated above, I encountered error, complaining that it can’t connect to database.
It stayed same even after tinkering with other wp-config values, and changed only after I changed SUBDOMAIN_INSTALL to true – then the error changed to TOO_MANY_REDIRECTS (why so?).
I am unsure if SUBDOMAIN_INSTALL is meant for this situation or if it is intended for naming sites within the network as subdomains. TLDR

Then i tried to do fresh install of wordpress, which worked out of the box, accessible even from our subdomain (localization.exe.sk), but after enabling multisite network (by following guide on wordpress, what had worked for me before, just not on the production server), it had stopped working, with same errors as the new page before – SUBDOMAIN_INSTALL=false -> DB error, SUBDOMAIN_INSTALL=true -> TOO_MANY_REDIRECTS error.

The testing server is accessible only from within the domain or via VPN.
The production server is accessible both worldwide and from within the domain/VPN (the worldwide IP address is different from the internal one).
It seems to be primarily a Multisite (which is crucial for multilanguage functioning) issue because disabling Multisite allows everything, including the database to work correctly.

Server Configuration with httpd:

<VirtualHost xxx.xx.x.x:80>
    ServerName 172.16.5.5 
    DocumentRoot "/var/www/exe"
</VirtualHost>

<VirtualHost xxx.xx.x.x:80>
    ServerName localization.exe.sk
    DocumentRoot "/var/www/exe"
</VirtualHost>

The xxx placeholder represents the internal IP address accessible only through VPN/domain.

PHP Version: 7.4.30.

Multisite Configuration in wp-config.php:

define( 'ALLOW_MULTISITE', true );
define( 'MULTISITE', true );
define( 'SUBDOMAIN_INSTALL', false );
define( 'DOMAIN_CURRENT_SITE', 'localization.exe.sk' );
define( 'PATH_CURRENT_SITE', '/' );
define( 'SITE_ID_CURRENT_SITE', 1 );
define( 'BLOG_ID_CURRENT_SITE', 1 );
define( 'FS_METHOD', 'direct' );

Any help or guidance on resolving these issues would be greatly appreciated. Thank you!
At the moment, on the localization.exe.sk, there is just english page from the new version of page, with disabled multisite network.

The page I need help with: [log in to see the link]