The unofficial 4.1 ROM has been leaked, you might try flashing tHAT.
If the RasbeanJelly ROMs are available for S3 then I'd suggest you try those. I have tried AOKP, CNA, Cyanogenmod and RasbeanJelly, and RasbeanJelly kerplucks them all. It's the smoothest of them all. I always felt that AOKP and CNA were the smoothest but RasbeanJelly changed my views forever.
And I'm sure that if RasbeanJelly is available for S3 then even Trinity kernel would be. Try this combo, it's a beast and undoubtedly the best combo!