I suspect the poor rat is dead by now. I had one a while back. I’m not keen on rats, but felt very sorry for that one and found it quite upsetting. But there was nothing I could do, but leave it to die in peace.
Once, it’s dead, the best thing to do is to put it out with your refuse collection. (I was told to do that many years ago by our local council. So still follow that advice). Just hope it isn’t too long till your next collection. Unfortunately mine had just gone the day before so the poor dead rat was in the shed for nearly 2 weeks. I would double bag it. I eventually needed to triple bag the one here, because it started to smell even through 2 bags.
But best not to leave it lying around outside just in case something else tries to eat it. I used newspaper to pick it up as well as wearing gloves. So it was wrapped in newspaper inside the bags.