Lol! The image of Schroedinger getting L a dead rat is AMAZING. He would understand the sentiment and not be too horrified, especially with an understanding of Schro's origins and nature.
One thing I thought of was a chess set, if Schro likes chess. Since L is wary around non-humans he tends to be more cautiously diplomatic, and unless Schro told him himself that he wanted one , L probably would be afraid that a cat toy would offend him. ;)