The drink's inventor was Hainanese, though accounts of the drink's history and contents vary.
The only reference to it in any form of Chinese that I was able to find was apparently written in Mandarin, where it's called
新加坡司令 which clearly is a Mandarin transcription (
sīlìng) and clearly not a calque of any concept of "sling".
I can't help but wonder if the name
新加坡司令 is a recent invention from China; Mandarin was not a major dialect in Singapore's early history, and a
新加坡司令 sounds more like a Singaporean citizen's commanding officer during national military service than anything to do with a drink made in Singapore. Attempts to compare the name to other related names don't seem to be of much help; Singapore's ASEAN professional basketball team, the Singapore Slingers, are known as 新加坡騰飛之獅籃球隊 in Chinese, and
騰飛之獅 also has nothing to do with any concept of "sling".
What do Chinese-Singaporeans call it in their local dialects?