An old fashioned word or phrase that describes a meditative walk. It's not a constitutional but it's something that Ben Franklin, or Thomas Edison would do to clear their mind, rejuvenate, reconnect, invigorate, rebalance etc. I think there is a noun for this activity. They might say "I am going to take my ...."
Perhaps the word you are looking for is perambulation. The verb perambulate is defined as:-
To walk through.
To inspect (an area) on foot.
To walk about; roam or stroll.
so the third meaning turned into the noun form would be quite close to what you want.
Kinhin is the walking meditation that is practiced between long periods of the sitting meditation known as zazen.
I doubt Ben Franklin or Thomas Edison used this term, but there is the Australian word walkabout, which Merriam-Webster defines as:
a short period of wandering bush life engaged in by an Australian aborigine as an occasional interruption of regular work — often used in the phrase go walkabout
I suggest you take the word jaunt and attach an adjective of your choosing to it.
an invigorating jaunt
a meditative jaunt
a refreshing jaunt
a mind-clearing jaunt
a re-balancing jaunt
a re-centering jaunt
Jaunt, by itself, suggests a pleasurable walk or perambulation; by combining it with an adjective you conjoin pleasure with mental refreshment.
It's also called mindful walking and the context can be found here
Also consider 'I am going to take my ritual walk'.
You could replace walk with other suggested synonymous words.
It's also common to say refreshing walk as in a 'long and refreshing walk'.
The phrase "morning constitutional" brings victorian gentlemen walking in Hyde Park to my mind.