What is a Singapore Sling drink known as in the Chinese dialects of Singapore? 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?
Like many words characteristic of Singapore, it is most likely to be said in English/Singlish. 

Local Singaporean Chinese, whether Mandarin, Cantonese, Hokkien, Teochew, Hainanese, Hakka, Fuzhounese or Fuqingnese, does have a very large range of loan words related to the circumstances of British rule on the Malayan Peninsula. One of the most salient examples when in Singapore is the use of the English form of address uncle and auntie, which is cross-topolectal and trans-lingual, and by extension, racially neutral. A common example from Malay is mata for the police, which is strongly preferred over most loans/calques of 警察 by Singaporeans of the pioneer generation.

However, sling is even more likely to be an English loanword, as it was originally a "gin sling", a form of specifically Western alcohol served primarily to Western patrons in the colonial era. One reference from 1903 reported in the SCMP makes a comment about "pink slings for pale people". 

Though Boon was (likely) a native Hainanese speaker, his position as bartender at the Raffles would have meant he was in an Anglophone environment for work. The sling was already  an established drink in the 1910s in Singapore. Hence it was likely referred to and discussed in English, even in predominantly Chinese topolect conversations.