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All they would do is arrest you for spurious reasons, then they have the right to that information.
Here are the details for stop and search by the way from liberty. https://www.libertyhumanrights.org.uk/advice_information/stop-and-search/
Liberty (I believe) are the group that give out cards informing of your rights at demonstrations.
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Bustcards are usually from the G+BC, but Liberty are good people too.
I'm not sure the cops I've come in contact with ever felt like they couldn't find a way to search someone if they want. I'm almost always on the defensive (as is my right) and ask if I'm being detained and what powers are being used to enforce this, they always, always have an answer. The easiest one is "well, you seem to be hiding something/nervous/uncompliant, this makes you suspicious to us, so it gives grounds for a search under section [cunty unrelated section of law that almost definitely doesn't apply]"
I honestly think that even if I had a copy of the section they're trying to stick me with and a solid defence against it, they'd still search me, and I'd still have no way to prevent it.