I know it has been debated over the years as to whether it might be problematic mixing kibble and raw....don't know if there is any merit to either side of the argument since I never mix the two. I feed straight raw for my dog's primary meals and then use kibble as treats throughout the day but the amount of kibble as treats probably amounts to 1/2 cup or so.
The throwing up bile when the dog is hungry does not sound common to me but perhaps others might suggest it is ???
Maybe try feeding the kibble and raw separately at different hours rather than mixing and see if anything changes. I also add items to my dog's raw diet such as yams, which have been put through the food processor until they are pureed. I also use pumpkin,eggs (whole) as you do...and at times add some yogurt or cottage cheese. I'm a big fan of incorporating green tripe as well.
I have always tried to make only one change ( addition or subtraction ) to my dog's meals at a time and then see the results so I don't have so much guess work if I make more than one change.