Thanks for all the suggestions & comments - it seems like there's no really clear answer but there are some opportunities here.

My return flight from GRU is next week so I will report back on what happens. I'll start with the LATAM lounge and perhaps then try the one that is provided to KLM Y flyers. I'm using the Delta app and I've seen the Delta Gold Medallion tag listed there before as an equivalent for my STE+/KL Gold status, so I hope that'll convince the dragons

I also wonder whether the GOL lounge will provide access since I'm not flying KL, and whether it's even worth it to make the trip to T2 and back to T3 for a mediocre lounge experience.