It's pretty clear to me that a lot of the gluten intolerance being claimed by tons of people is basically self diagnosed fad following. I'm NOT saying gluten intolerance is not a real thing that truly affects real people, but without a medical diagnosis, a lot of people don't know a hang nail from a runny nose.
Food allergies are definitely a whole different thing. A allergic reaction is characterized by production of histamines in the body. Histamines are messaging chemicals sent out by your immune system which cause your skin to break out in hives, your lips to swell, runny nose, itchy eyes, etc., and in extreme cases they can cause anaphylaxis where your airways constrict to the point of preventing respiration plus low blood pressure.
Having indigestion after eating a BLT on whole wheat is not technically an allergy.
The OPs symptoms do suggest a possible allergic reaction to me. What happens if you eat a spoonful of wheat germ by itself? How about if you rub a little into your skin?
Other foods that are commonly used in smoothies are also common allergens. Strawberries, apples, cherries, etc.