Adamo,
It is good that you have a doctor. Accept his help.

Since this is what the doctor suggested, and you have only been on Klonopin for a few weeks, I'd try that.
I used a benzo for about 3 weeks inpatient... was given a lot of it plus ambien plus another sleep benzo. When I came out I was told to use it as needed. I only used it maybe once a week or twice and was fine. Did not notice a darn thing. The first time I ever tapered was much later in my life, after having been on a benzo daily for 6 years.
It varies by individual but generally someone who has used it less than 6 months or so does not encounter protracted withdrawal. It is not even all that common with long-term (years of treatment) patients.
The cortisol thing applies only if you are exercising at your max aerobic rate... on the verge of anaerobic. Keep it below that. Also, you might invest in a heart rate monitor if you exercise at those levels.
The overall cortisol picture is very complicated because that hormone goes up and down with so many factors. One thing is for sure though. Cortisol goes up with stress. This situation is stressing you out. Exercise helps reduce symptoms of stress, and as such will reduce stress-related cortisol. In that sense, a moderate, sensible exercise program will help you.
As for the panic attacks, the best ticket ever to having a panic attack is fearing one. I can't say how common it is with d/t exercise during benzo discontinuation, but if you have the fear of it from the get go, do what you need to do not to be afraid... perhaps make your aerobic workouts less intensive than usual.
I think it would benefit you to stop reading horror stories. Working up fear is not useful here, it only sets you up for a negative outcome... it can be a self-fulfilling prophecy especially where anxious symptoms are concerned. Remember too, a lot of those stories are bound to be about people in very different situations.... i.e. long-term benzo patients.
Set yourself up for a positive outcome instead. Keep your focus on the desire to get your life back. If you want to research things, here is a suggestion. Research non-pharmacological ways to reduce worry and anxiety. For starters, a very simple, very effective thing to do is abdominal breathing. You have to practice it, but it mechanically forces your body to relax and the sensations will feed back to your brain.
I ran into some useful stuff of this nature a while back but I have to find it again... it will be for next time.
All the best to you. Try to take it easy... (I know it is easier said than done, but we must reach...

) I'll keep checking in with you. You won't be alone.
waves