We had this exact problem a year or so ago.
I put poison down in a box, and also in the other end of the tunnel iyswim - so nowhere near the chickies, in case they ate some by mistake.
The rats stayed around for a while and then a massive one ate all the poison in the box, and died next to it, which seems to have put all the others off! I haven't seen one in ages now.
Tunnels are still there but like you, we have slabs round the edges, tunnels are too small for a fox to get through and also too long (the main one I think comes from the other side of the shed)
I have left the inside part as plain earth, which is what they had to start with - I've never bothered with extra bedding in there, except some straw sometimes, as the roof is mainly covered in corrugated sheeting so it stays nice and dusty and dry, and the droppings seem to disappear into the ground without picking or cleaning out. (shed itself a different matter!!)
Is your run covered? If not then yes paving slabs would be fine I think but they wll need something over it to scratch around in - so earth on top, or something else - I'm not sure about concrete mix as I've never used it, and I think it might need a level solid base to go on top of?
Good luck.