I'm allergic to at food coloring, but I don't know how strongly allergic. Can this be tested without a scratch test?

Your body's clues. Your body has already given you some good clues. What reaction did you have? A mild itchy rash? Lots of hives? You know you're strongly allergic if you've had major hives, and especially if you've noticed swelling in your mouth, trouble swallowing or breathing. Talk with your doc about your reaction & what to do next.
Trial under MD care. You may be able to ascertain by challenging yourself with the suspect agents, under the care of an allergy specialist. I wish to emphasize that you should not try to it by yourself since some of the reactions can be life threatening.