score:2
Answering my own question since I just wrapped up both internships. I was able to apply for, and get, the Swiss visa from the Swiss Embassy in Singapore since I was a Singaporean resident at that time (since I had the TEP).
There were no hassles getting it but I would recommend apply early and like @JonathanReez mentioned, it's a good idea to email the embassy in question before proceeding.
Upvote:1
The Swiss embassy in Singapore requires the following documents to prove residency:
Proof of legal residence in Singapore, valid for at least 3 months beyond the intended date of departure from the Schengen territory. Original plus 1 photocopy back and front of Employment Pass / Permanent Resident card and re-entry permit / Dependent Pass / Student Pass.
So the answer is yes, as long as your Employment Pass is valid for 3 months beyond your intended date of return to Singapore.