Potential bingo: acne is actually as much inflammatory condition as it is an infective one. You definitely should be checking ingredients and again review your diet - plenty can be done to balance hormones, moderate the immune response and reduce systemic inflammation.
No, acne does not necessarily preclude co-washing with any product for anyone with acne, plenty of people successfully use the Oil Cleansing Method, but using a decent emulsifier and not over stimulating works best. I definitely would not choose products which are highly moisturising or contain lots of plant extracts, simple and gentle, acidic pH and rinses clean.
Comedogenicity is far from the only issue, known irritants and ingredients that damage the skin barrier/ acid mantle (keeps bacteria out) are just as much of a risk. Some with acne respond really well to dropping sulphates and other anionic surfactants, some worsen before they improve as the skin flora recovers and the skin becomes better hydrated.