Mine do get a whole leaf. That's how I used to feed out heads of lettuce. Stand in the middle of the cages and count pigs every time I tore a leaf off. 1 leaf for that pig, 2 pigs, 3 pigs, next cage, 4 pigs, 5 pigs, 6 pigs.... Oh look the lettuce is gone now on to peppers.
So long as you bought the lettuce fresh it will survive a few days chopped up. I cut all my veggies up weekly into plastic containers and then left the lids cracked so all anyone had to do was dump in a container. It lasted the week chopped up so long as the containers weren't air tight. In a plastic bag it won't last quite as long but would do for a few days. The paper towel will help if the lettuce is wet from either rinsing or just from how much they overspray things in the stores. Overwet veggies don't last as long.