One trick you could use is adding a parameter to the links where you call the CSS file with the version number. So instead of calling style.css you call style.css?version=1.1. That way you call the same file but with a new parameter which hasn't been cached yet. That forces the latest edits to directly show. Every time you make an update to the CSS, also increment the version number where you call the file. And if you also add the version number as a comment to the top line on your CSS file, then troubleshooting these issues will become much easier.
So in your LP template call:
<link rel="stylesheet" href="[...]/lp-style.css?ver=1.1">
Hope it helps!
Jerry
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