Drowning animals is illegal in a lot of states. Nuisance or not
I know many states forbid drowning of 'domestic animals' as a form of euthanizing... But, when it comes to trapped nuisance animals most states laws (that I have I read) are silent on any approved or disapproved method of killing the trapped animals once trapped, or say something like 'humanely euthanized' and any interpretation is rarely defined thus the definition is up for grabs as you will find many on both sides of pretty much any euthanization technique including drowning... Also talk to any live long fur trappers, drowning is pretty much the go to way they do it as to not damage the hide and is approved in many states wildlife hunting/trapping regulations for trapped animals and is even schooled in many hunter safety classes when they discuss trapping...
Now of course laws will vary, and in many municipalities there are a slew of laws that can change from town to town, many even forbidding the trapping at all by unlicensed individuals... Heck in many areas around me even BB/pellet guns are illegal to discharge outdoors withing city limits, let along using a real firearm to dispose of a trapped animal...