I would recommend writing the plural of the name "Kos" as "Koses". The answers to the related questions
Plural of 'yes' and What is the preferred plural form of “bus”? indicate that there is no general requirement in English to double "s" when it comes after a short vowel and before the plural ending "-es" (in fact, the spellings yeses and buses seem to be preferred currently).
The answers to How do I pluralize a name ending in “y” indicate that there is a general reluctance to alter the spelling of a base proper noun when the plural affix is added. From Alex B's answer:
Huddleston and Pullum 2002 give the following rule for proper nouns: “the base always remains unchanged in both speech and writing” (p. 1595).
That said, it is possible to find examples of people doubling s to ss in the plural forms of other, more common names. Merriam-Webster says the plural of "Gloomy Gus" can be either "Gloomy Guses" or "Gloomy Gusses". So I think you may have some choice in this matter.
English spelling is often ambiguous, so while I understand the desire to indicate the pronunciation more clearly, I'm not sure that's a very convincing reason to use the spelling "Kosses". It's true that "Koses" is ambiguous, but even "Kosses" would have some ambiguity of its own: there's no way to know for sure that it rhymes with "mosses" instead of rhyming with "posses".