@Gwenhwyfar
"Also, if it isn't living in there but coming in just sometimes then my biggest concern would be a cat or fox finding it and eating it and dying themselves."
The poison can be in containers that only mice would fit into.
The problem is that most poisons are slower acting so the pest moves away from the bait and dies elsewhere, allowing more pests to come to nibble the bait without the big red flag that is a dead compatriot lying next to the food source.
Because it's slower acting allowing the pest to go elsewhere that means they can make their way back outside and either die from poison and be eaten by carrion eaters or be caught by someones cat/dog or an owl as a PP pointed out.
Introducing poison may take care of your immediate rodent problem but it also introduces poison to the wider ecosystem.
OP if the neighbours don't have their house rodent proof they will already have visitors, you won't be causing a mass exodus of rodents that suddenly take up residence with your neighbour if you put out deterrent plug ins, honest.
If you're worried let them know you've got rodent sign in the house and you are taking measures, maybe they should have a check too?
Don't get the dogs in, that's just as horrific a death as drowning
unless there's masses of them or they are in a confined but open area dogs won't get many anyway, they'll just hide.
Agree sticky traps are awful torture devices, the poor things just chew off limbs to escape 
Prevention is better than cure but if you must use fatal methods, please use the quickest and most humane ways to do it. A quick bop on the head with something heavy, snap traps or shooting for rats if the shooter is an accurate shot. I prefer humane catch and release but as a country resident for most of my life sometimes popping them off is the only way. You must do it as kindly as possible please though.
You cannot have wild rodents running amok in the house, they do incredible damage to things and can make people ill from their urine which will be everywhere they have been. Anyone suggesting you allow them safe continuous living in your house is some nutty Cinderella wannabe who doesn't know what real rodents are like - and I love rodents! 