Hi there,
Quick question: Will your chickens have 24/7 access to said flowerbed?
If they do, it is relatively difficult to keep plants they like alive, if the chickens have access to the whole plant.
My chickens really enjoy grape leaves. I used to have a couple great big grape vines and any vine that didn't attach at the trellis or up higher and flopped down would get stripped by them, as much as the chickens could get to. You'd be able to see how high the chickens could get the leaves. One time we trimmed a bunch that grew into a tree and put it all in their coop and when we put them away, they stripped those too.
Clover is always a good basic one.
Mine like rose petals. I learned this after trimming old roses and throwing the petals for them. Then one time a rose leaned over and they got some of its petals. Funny thing to come across haha
As far as supplements to diet, having oyster shells or some other form of calcium available is a good idea. Hens will eat however much they want.
Hope this helps!