As you conclude from the responses, not many agencies have specific policies on how to euthanize an injured animal. In my experience, policies usually authorize euthanization, but do not go into specifics. Practice is usually dictated by training and custom. This is likely different in some agencies that have to euthanize more frequently or have had an issue with a particular shooting. (If I still worked for my former department, I suspect I'd be sending annoying memoranda asking internal affairs guidance on when I could use what weapons on what animals and advising that I would be calling the internal affairs lieutenant, on or off duty and any any hour of the day or night, to seek his wisdom and guidance. He should have some interest in this being the little rodent that he is. That said...)
In my experience, officers are cautious about shooting injured animals. While no one wants to see an animal suffer for longer than necessary, one has time to carefully evaluate the scene to avoid collateral damage. Officers are cognizant of the risks you mention. Officers may clear scenes or block traffic. Unlike response to a deadly threat by a suspect (or even an aggressive animal), one seldom has to shoot in a split second decision when downing an animal. In most cases, the round or rounds could be fired in a downward direction reducing the risk of passthroughs.