Just a rat? Pain and suffering is pain and suffering.
When I had a rat problem in a coop a couple of years ago, I set a hav-a-heart trap. Next morning, a baby rat was in the trap but nestled in a perfect nest of straw. It's parents must have fed straw through to him/her as the trap started out with no straw in it. So while we may find them problematic they care for each other in ways not that different from any other species.
I would opt for the quickest and most humane kill possible if killing is the goal.
In my case, we finally figured out how to rat proof so well that this ultimately solved the problem. They took off for greener pastures and never came back. They had been making quite a mess so it was a frustrating situation - I understand why people want them gone - so did I. I could barely get out the door to work with all the clean up I had to do each morning.
The rat proofing was for me better than the hav-a-heart idea that I started with (essentially to ID what I was dealing with because I wasn't sure at first) because once caught, then what. Didn't want to transfer the problem to anyone else. I also tried purchasing repellents that claimed great success but they didn't work at all. Re: poison - I didn't want to risk poisoning unintended victims - have seen that happen and it's a bad scene; a much loved cat of mine took very ill soon after a neighbor put out rat poison - the cat did not make it. There are posions that won't affect cats but some of them will kill other unintended species like bunnies and squirrels. If using poison get the one that will affect the least number of species and locate it where these other species cannot access it. There is still a risk because rats move their food around, and they do the same with their poison.The only fail safe method is to put it where only rats can access and have a system where once the rats go in they can't get out with the stuff. I have seen postings on BYC where innovative people have figured out just how to do this.
Hardware cloth was the saving grace on rat proofing - they couldn't get through it and they gave up and went away. We installed it over the insulated walls and even on the wooden floor (which was then covered with straw)
JJ