This post probably be entitled "Plesk Install Guide without SSH access" or similar as with SSH access or access via the new terminal app from plesk it's much easier, granted not straight forward - perhaps the hardest parts are...
I've tried using the composer update tool but it's not reliable in my opinion I often see things already updated, needing to be updated.
If you happen to know or can get in touch with the server administrator then asking them to sort out composer 2 support will be a massive benefit - it's officially supported by Plesk now (support article)
but it only really needs the below in the panel.ini and a quick update of the phar file, the biggest reason to dump composer 1 is it's horrid abuse of memory compared to composer 2 (at least on Plesk)
composerVersion = '2'