Chopping it up will likely cause it to wilt faster. Also, when you cut lettuce it get’s that reddish brown oxidization. Not sure if those bit are ok for rabbits to eat but I always end up picking those bits off.
Keeping it on the core will keep it fresh longer. I usually cover lettuce and greens in slilghtly damp paper towel or cloth but making sure the greens themselves aren’t overly wet. Wet greens go slimy. Don’t toss the core either. Trim of the yucky bits and give it to Lolli to see if she likes it. Jersey does, Rumball doesn’t.
Lolli sounds like she has a taste for the sweeter greens. Full grown romaine can taste more bitter. But if she eats it eventually then good. I wouldn’t cater to her sweet tooth too much. Mine will eat their faves, sweeter stuff first but enjoy the other greens if that is all that’s on the plate.
I tend to buy curly endive and radachio a bit. Not really lettuces. I find they’re a bit more robust greens and keep better. Also, give brussel spouts a go. Mine love them and they can have one each without any issue with gas.
I’ll provide pictures because of geographical difference in what we all call different veggies.


