It does have a greenish cambium guessing because of the chlorophyll production in the bark.
For the most part 'bark' is dead tissue. The chlorophyll is in living cells such as bark cambium (on the outside of the phloem tubes) and the vascular cambium (that we simply call 'the cambium'). There is also parenchyma distributed through the wood. I acer palmatum, young stem wood is green as is the cambium and the bark looks green. All of this photosynthesizes, but it is minor compared to the foliage. Like
@grouper52, I had no idea so much of photosynthesis in Palo Verde occurred outside of the foliage.
I've never really pursued the question of why (vascular) cambium in most trees is green, even when the bark is very thick. Clearly it isn't photosynthesizing when the bark is very thick and opaque, yet the cambium has chlorphyll - what else does chlorophyll do besides photosythesis?
As to foliar feeding, the phloem tubes are in the leaf, so anything that gets into the leaf, through the stomata or otherwise, has the potential to get into the phloem tubes and to trickle on down the tree in them, getting unloaded by the daughter cells and into other living cells somewhere below. It was a physicist (Fermi?) that applied something with a radio tracer element to the leaves of the plant and then detected the radioactivity far away, down the plant, to prove the plausibility of foliar feeding.
