This is the other one we used effectively. Takes large batteries. You smear a bit of chocolate spread at the end (chocolate spread works best) and turn it on before going to bed. The mice are instantly zapped when they walk in so it's fairly humane. It can also catch quite a few as it's big enough to catch a rat. You don't have to look at or handle them - just tip it out.
https://www.amazon.co.uk/Pest-Stop-P...8442490&sr=8-2
If you do decide to catch them in a bucket or humane mouse trap you need to set them free at least a mile from your house or they come back! Homing thing. I'm afraid I couldn't handle doing that at 6 am every morning - taking mice for a ride for a mile.
You just need to be methodical about it for a few days and keep the cats and hamster away. But if you really think it's too risky there are these. I didn't like them - the mice also got clever and learned how to get the bait without getting caught! I'm sure one of them held the door open while the other got the bait - but they got wise to them. Those are the ones where they can die of fright if not let out of them soon enough (hence getting up at 6am and driving a mile to let them out) and you may have a few to transport!
https://www.amazon.co.uk/Pest-Stop-P...8442824&sr=8-3
If you want to try the bucket trap - use one of those tall kitchen bins instead of a bucket. They can't jump out of that as it's too high.
Personally I'd get the sealed rentokil traps and leave them down a week. But if you're worried about the cats finding a dead one before you do and eating it, then I'd go with the electronic one.