Hair of the dog would not contain anything that would lead to constipation- if anything- the opposite as omega vitamins aid in digestion.
Soaking the kibble would help. I also do not know what brand of food you feed.
My guys like string beans over green, but carrots is still the prefered. What you could do is steam the green beans or string beans- use the water as the water for the kibble.. The problem with cooking any veggies is that you do lose half the b complex vitamins in the veggie.
Which brand do you feed?