Again, if page is cached PHP can't be executed and you need to solve it with .htaccess Rewrite Rules. .htaccess has no logic to detect if client accept ccokies or not, so you can only check with with .htaccess if a specific cookie exist or not. If not -> no-cache. Otherwise such clients like...
For Virtualmin is no plugin available, so you must do a Generic LSWS Web Server Installation, but be carefull. It is not tested:
https://www.litespeedtech.com/support/wiki/doku.php/litespeed_wiki:installation:generic
Of Course there is a way, but any other way brings new issues. The consequence of this dilemma could be to disable cache for any users that doesn't accept cookies.
RewriteCond %{HTTP_COOKIE} !name_of_vary_cookie [NC]
RewriteRule .* - [E=Cache-Control:no-cache]
Not lscache uses cookies, your plugin uses vary cookie. Using cookie to store information which device is used is a absolute wrong method. To me, bad idea to use cookie for this. Ask LiteSpeed staff if they re-design this plugin.
Strange.... Have already asked support of PHP framework? You know that you can set such settings also in cPanel for each account?
btw. you can disable zlib.output_compression. Set compression in LSWS WebAdmin to get zip AND brotli compression instead.
It seems you need a more powerful hosting. If pages are cached speed is okay, but if not cached your shop needs more than your server is able. You need at least a VPS with SSD hardisk.
Check your settings. Disable recompile template files if enabled, maybe that helps.