We had mice in our house out on the country side. we put out poison and the numbers was greatly reduced, I do not know how many mice that past away, but all except one choose to pass away outside as they are searching for water. The one who choose to end it's day inside the house made a rather unpleasant smell, but it disappeared after just a few days. If you can't catch in a trapp then I would use poison most likely it will die outside else you will have a smell for probably a few days more than we had as a rat is bigger animal.
Rats do like beer, if you put a sizable amount of bear in a bucket, and then put some items that floats on top of the beer, I have seen it made with ordinary wine corks but anything that floats will work, such as foam from a yoga math. They can't climb out of the bucket if the bucket is high enough and drown rather quickly. It can be worth to offer some beer for a try. You can also put some kling film on top of the bucket with a hole in the middle that would also add to the complexity for the rat to escape.
Paul on S/Y Kerpa SM#259