Unwelcome guest?

As you must realise, I’m a bit of a wildlife geek, but I have a bit of a problem, no, not the pike in the pond, but a rat under the shed.
Living on the edge of the countryside, which is literally over the fence, I’m not surprised that creatures tend to move in.
We have a lovely fox wandering about, plus deer and the birdlife is amazing.
Last year Mr Rat decided to have an easy life and live under the shed where he can sneak out and feed off the seeds that have fallen off the bird table.
Thing is, he is massive, but not one of these tatty old urban scumbags, very sleek and handsome, good advertisement for the perfect rat.
My dogs try and catch him, but they can’t, our Jack Russell spends much of his time barking under the shed, to no avail.
Unfortunately I realise that rat life will accelerate and we could be over run, but I can’t put down poison, it’s evil stuff and he doesn’t deserve it, neither could I possibly contemplate getting out the air rifle either.
Any ideas as to how to remove him, personally, I think he’s very brave to hang about here with a Jack Russell about, old Scooby Doo probably keeps Ratty on his toes and prevents his mates from moving in, I have to admire the little sods cheek, as he’s not doing any harm, and with the constant dog threat, perhaps I should leave him to it, everything deserves a good life, doesn’t it, like it or not, what do you think.