I would think that the smallest lightest varmint bullet would do the trick (Maybe a lightweight .223 load - 45 grain V-max maybe?) . You are looking for something that expands immediately in a very small light target. You might want to try slowing down your loads. At those ranges, you don't need all that speed and a slower speed might give the bullet more time to expand/explode. Not sure, just thinking out loud. I am sure that more experienced varminters will provide you with a good answer.
If I may be so bold as to translate for the rest of my Mates in the USA: Esky is short for Eskimo - a brand of cooler - probably looks something like this Igloo cooler, except with footprints on the top:
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