This issue isn’t directly related to LiteSpeed, but more so the way I have WordPress set-up, sorry about that!
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The problem was the WordPress absolute path is set to /wordpress/ rather than /sites/<site>/wp.
I had /wordpress/wp-config.php working as a proxy to load /sites/<site>/wp-config.php. The WordPress absolute path has already been set at this point by /wordpress/wp-load.php.
The solution is to set the ABSPATH before WordPress loads, by setting an auto-prepend-file in .htaccess.
https://www.php.net/manual/en/ini.core.php#ini.auto-prepend-file
https://www.php.net/manual/en/configuration.changes.php
