Would taking 100mg doxycycline and a probiotic regularly be an issue?

Possibly: Doxycycline is a broad spectrum antibiotic which has the capacity to alter your normal bowel flora (types of bacteria) and to promote the outgrowth of other organisms. Clostridium difficile colitis can be a life-threatening disease. The probiotic may help to prevent this, but many of the organisms in probiotics will be killed by Doxycycline so it may be a double-edged sword. Hard to predict.