Uhhh So sorry to hear about all these digestive issues in your pup. No expert but maybe a few suggestions I have from experience . Find ONE quality food to feed. I go by this sites 5*. I believe the brands/proteins which are 5* has less fat to cause digestive issues. Then start feeding a FEW kibbles at a time at MANY times of day to keep stomach full and to get USED to kibble. SLOWLY increase. A LITTLE plain boiled WHITE MEAT only chicken breast with NO broth added. Maybe a string bean or two. Bile vomit is usually from empty stomach.
The trick is MANY SMALL meals several times a day. But introduce the new kibble SLOWLY. New food will usually cause loose stools at first. But switching again will start the digestive problem over again. So give the new kibble a chance. Just go SLOWLY. If after a week or so he does well then divide feeding to three times a day. Stick to only the kibble.
After several weeks you can add in other food as topper. Tiny pieces of lean steak, again boiled WHITE MEAT chicken breasts, canned no salt string bean or two. Keep the tummy full.
Just stick to ONE kibble in the begin and give a chance to get used to. Hope I was of help a little .
