Is Germany actually attacking your ships in the strait, or is it bombing Denmark? CAG Duty will only function if one of your fleets in the province is the target of the attack. Bombers can fly across that province with impunity.
More confusingly, bombers enroute to Denmark will never actually exist in your sea province, they teleport up to their maximum movement distance each hour and appear in the target province, never actually entering any of the provinces along the way. To intercept the planes heading to Denmark, you've got to Intercept them over the target, over their own airfield, or at whatever province they happen to make their hourly pause in along the way, if the distance is more than they can cover in an hour. If your fleet is in a province (or sea province) adjacent to the target, your planes may occasionally intercept them, but it's not very reliable.
More confusingly, bombers enroute to Denmark will never actually exist in your sea province, they teleport up to their maximum movement distance each hour and appear in the target province, never actually entering any of the provinces along the way. To intercept the planes heading to Denmark, you've got to Intercept them over the target, over their own airfield, or at whatever province they happen to make their hourly pause in along the way, if the distance is more than they can cover in an hour. If your fleet is in a province (or sea province) adjacent to the target, your planes may occasionally intercept them, but it's not very reliable.