Do not cache html element

1 way to exclude output from beeing cached is to use this cookie for cache vary. For example:

RewriteCond %{HTTP_COOKIE} cookiename [NC]
RewriteRule .* - [E=Cache-Control:no-cache]

If this rule should only be active on a specific URL:

RewriteCond %{HTTP_COOKIE} cookiename [NC]
RewriteCond %{REQUEST_URI} ^\/url
RewriteRule .* - [E=Cache-Control:no-cache]

you can also try use ESI block

please check this page

https://docs.litespeedtech.com/lscache/lscwp/api/#generate-esi-block-url

You can also consider to use JS to control that css class.