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I found a set of the Freiftaat Bayern overprints of the King Ludwig stamps with a K perfin on all of them. The set includes a 5, 10, 15, 20, 25, and 30 Pfennig stamps and all are mint hinged.
Those are likely not company perfins, but so-called "postage free" stamps (Portofreiheitmarken). From Michel, roughly translated:
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In 1919/1920, so-called postage-free stamps were introduced for offices whose postage fees had to be paid from public funds, but which were not allowed to use normal official stamps due to bad experiences (ministerial decrees of February and June 1919). In order to be able to control consumption, circulating postage stamps were initially to be overprinted, but ultimately it was decided to perforate them for reasons of economy. Their use was soon stopped, however, because the stamps were being misappropriated for collecting purposes.
Those issues were for use by War Disabled Persons' Committees (Kriegsbeschädigten-Ausschüsse). Here's the listing:
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Michel also notes that there are many forged perfins of these issues.