Rat King Catalyst is "Found in Strikes against the most challenging opponents" so no, you can't get it from the Crucible.
I don't know what the Clan Banner perks are off the top of my head, but sometimes there's a level perk where yes, there is a higher chance when playing with Clanmates (check the Clan Banner for that information). There is no increased chance by playing with other Rat King users.
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Oh ok cool. I was just making sure. I appreciate the heads up