You seem mistaken on the role of cortisol in the body. I realise there is a lot of confusing information out there that can lead to such a misunderstanding.

Cortisol does not "cause stress".

Cortisol is a metabolic hormone, it's primary function is to increase blood sugar through gluconeogenesis in the liver. The increase in cortisol while doing exercise is quite healthy and beneficial as it means the individual won't suddenly have to stop due to low blood sugar.

The reason cortisol is sometimes referred to a stress-hormone is that it is also a feed-forward hormone - in two ways. The first is that there is a daily cycle - it is low while we sleep and spikes in the morning to give us a boost, to deal with the spike of activity in the morning when we get up. Secondly the level of cortisol in circulation will increase when an individual anticipates a need for higher activity levels (and thus metabolic demand) in the near future. If the individual actually needed that spike in blood sugar due to increased activity then this is quite beneficial (at least in the short term).

However, this increased cortisol can sometimes lead to problems (including weight gain and risk of type 2 diabetes) if an individual constantly anticipates or experiences problems ("stress") over long periods of time without cessation.