Does running Croct plus the Clarity integration slow the page down?

The integration does not add a client script from Croct at all. It is outbound server-side data streaming, Croct sends the audience, experiment and variant dimensions to your Clarity project from the backend. So there is no new bundle downloaded and nothing extra scheduled on the main thread by the integration itself.

As for Croct's tracking generally, it can track several events within 2 to 3 milliseconds, which is negligible against an LCP budget. The thread on what event tracking costs on the main thread breaks that down if you want the detail.

Adding the rendering side, because that is usually where the LCP damage comes from with client-side tools: content resolves server-side, so there is no anti-flicker snippet and nothing render-blocking waiting to swap the page. That is the opposite of the prehide approach you get with the older visual testing tools.

Your existing Clarity recording script is separate and completely unchanged by the integration, so its cost is whatever it was before, no better no worse. If you want the full picture of what Croct's latency claim covers, what the sub-90ms P95 number actually includes is a good reference.