Why would you worry about that? You can exercise to exhaustion without worrying about negative effects of cortisol. The exercise releases many other chemicals that are helpful. It increases memory and certainly reduced the chances of brain damage of stroke. In aids bone growth etc.
While you can have problems with asthma you can also reduce the chances of type 2 diabetes.
In short - the advantages of exercise of any sort are far more positive than negative if not taken to extremes. And cycling is the best form of exercise for people over 40 since it has far less stress on bone, body and brain.