When you say "the corporate ladder", I suspect your thinking might be too constrained.
Certainly many businesses want leaders that devote significant time and energy to the company. Certainly some business involves sedentary work. Some businesses are inside. And many businesses are stressful. But not all!
There are "ladders" which involve physical activity, time outdoors, and perhaps low stress. If these are important factors in your life (and it sounds like you believe they are), then you need to put yourself in a position to maximize those attributes in your work life.
And then there are folks who are self-employed. They get to decide for themselves what their work life is like. They can craft their work to meet their individual needs exactly.
I have never told my company about my condition, out of fear that it
might sound like an excuse to not overachieve, but I am considering
whether I should, so they might be more lenient in their expectations
I'm not sure what you are expecting your company to do here. Are you expecting that they will say "Oh, you have this condition? Then, we'll let you climb the ladder without working as hard as others on the same ladder."? While many companies can easily accommodate all sorts of individual needs, that doesn't mean everything will be equal career-wise. What you deliver for the company is typically most important. If you can still deliver top performance despite your condition, then you will likely be in good shape. If you cannot (for whatever reason), then you can't expect to be rewarded as if you could.
That doesn't mean just because another manager works 60 hours per week, that you must also work 60 hours. Perhaps you can perform equally well on just 45 hours (because you work smarter). There are many ways to achieve your results.
Can I (and if yes, how) continue climbing the corporate ladder without
sacrificing so much time?
It sounds like, in your current company and in your current position, you haven't yet found a way to do so. So, while there might be ways (perhaps by adjusting your definitions of "continue climbing" and "so much time"), it isn't clear that you can.
In my current company, at a certain level, you are expected to sacrifice a lot of your personal life for the company needs. For example, some of the senior folks spend the entire week away from their families, and only return home on the weekend. That's not something I would ever do, but they are rewarded well, and it seems to work out well for their families. Everyone is different.
Perhaps if you like your company, you need to re-think your goals a bit. All of us eventually settle into a work-life balance that meets our needs. Some of us work more, some of us work less. Some accept a lot of stress, others not so much. The key is to get to a place where you are happy with what you are doing and the rewards you get back.
Don't feel that you need to be the top "ladder climber" at your company. Perhaps you are better off climbing less or climbing more slowly, and receiving better personal relationships and health in return.
Or perhaps you need to find a different situation that better balances your work-life needs. Only you can decide.