Just to complete the loop... SQ check-in agent in Sydney was indeed more than happy to interline my checked-in luggage onto my (separate ticket/PNR) KLM flight onward from Amsterdam. She had to make a quick phone call before checking me in, took about 5mins total. Could only get the SQ boarding passes, not the KLM one.
Having done this, I was also unable to OLCI for the KLM flight using wi-fi at Changi (the KLM flight check-in having opened up by then). It gave an error message saying to check in at the airport.
Picked up the KLM boarding pass successfully at AMS transfer desk. Not sure how it would have worked if SIN-AMS had been late, but I guess that would have been my risk of booking the 2 tickets separately (though not sure what would have happened to my bag in that case!)
Thanks to those who replied to help previously, it all went seamlessly... Well done SQ for interlining outside alliance on separate tickets (where many airlines won't these days!)