I tested positive on hydrogen breath tests for lactose, fructose, and lactulose. What additional tests should I request? I still feel sick.

DON'T. You do not request tests, you discuss with your doctor or doctors and together come up with a plan. If not satisfied change doctors. Just a tip when you go to a doctor and ask for a test to be done some doctors get a little put off. Believe me it is not just me. I would send you on your way- immediately. Lactulose is a medication.
All this means... ...Is that you should not consume foods containing large amounts of lactose or fructose (lactulose is not a food sugar.) if you feel sick while avoiding those, then the problem is unrelated to this intolerance.