Talking theoretically and maybe assuming you happened to be on holiday abroad when SHTF struck :
Depending on your experience of larger animal hunting, you might want to consider some training/study/course in tracking, as a suplementary prep. My thinking is that a slingshot would have limited capability to create an exit wound and with the projectile still plugging the entry wound, the opportunity for a decent blood trail could be limited.
Brave guys who hunt brown bear with one