Hi again,
You don't say what those various quirks and issues are you're experiencing with the Litespeed Cache plugin for Joomla. However, it appears that it's being maintained (see
github.com/litespeedtech/lscache-joomla
-- last updated 4 days ago),
Also, the plugin requires proper setup (including at least one modification of .htaccess), and yours is not known, either.
Please note: I'm not saying "it's your fault" or so.
However, it's safe to say that the issues you're facing are not caused by HikaShop, either. It's all too common that, by default, certain server caches result in shopping carts and other dynamic content not being updated... without special cache control (such as possible with Litespeed cache and the Joomla plugin for it) you'd have the same, or at least similar, problems with Virtuemart or other shopping carts, certain functionality of CCKs and similar components as well.
About your case: obviously, you're dealing with a bottleneck somewhere. If your hosting company is recommending to engage their server cache, it's typically for one of these two reasons:
(1) You just want to "generally improve pageload speed", or
(2) you are hitting server resource limits set by your hosting package and are looking for tools or tricks to be able to stay within those limits.
Server cache can definitely help with (1), but for (2) it will be only a bandaid, not fixing the problem at its root.
In any case, if you want or need to get the Litespeed (= server) cache to work, you'll likely need to spend more time on getting the plugin set up. Unfortunately, there is no "one for all" or "plug & play" solution for this; it takes learning and eventually some tinkering.
If it's (2): to avoid the root cause, you may want to consider hosting that "busy eCommerce site" on a more performant server with enough headroom for the task, plus then having cache options for additional gains in speed.
But now I went really far off topic here (sorry for that -- just trying to help!) as, again, none of this is caused by or directly related to HikaShop, indeed.