For some reason the answer
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"Deportment" is correct, but for 99.9% of English-speakers it is an archaic, "intellectual" fancy term.
(Indeed - 70% of English speakers would assume "deportment" is one of those "fancy French words" that intellectual posh people use, like "bon appetit" or "say lah vee" or "sauce".)
Given that "deportment" is way outside the general intellectual / vocabulary range, indeed "poise" is a great choice - "Jane has poise" has the exact sense the OP wants.
Note too that simply "bearing" works.
(You often hear "military bearing", Steve has military bearing.
And there's an idiomatic phrase, the way he holds himself, and also the way he carries himself.
I appreciate OP is actually asking for the "art of..." the system and effort towards poise, bearing.
This concept literally does not exist in the anglophone sphere.
(Even the concept of "éleve", raise a child, is completely different. In French it means steward, herd, form - in the anglophone sphere it means "the party who pays for rent and food while the human moves from 1 yr to 18 yrs of age". You can't really translate "élever un enfant" to "raise a child", you have to go to "pastoral care" or "formed the man..." or such.)
Exactly as the OP says,
I am thinking about composure, bearing, poise, but these sound to me like qualities rather than crafts. Does it not sound weird to take lessons in bearing or lessons in poise ?
The nearest concept in English is "finishing school". In stories when a typically 17 yr old female is "sent to finishing school" (which would be either in (a) Paris or (b) "Switzerland") she is taught how to "walk with a book on her head" (you know?) which is the closest concept in the anglophone sphere.
In the deep South in the US, there are still a few clubs, courses, usually labelled something like "Miss Jane's Manners for Young Ladies" or indeed using the word "deportment" (recalling there is still some Frenchish language kicking about in the South) ... "Miss Jane's deportment classes". Instead of say a dance class or soccer on Saturday mornings, youngsters will go to Miss Jane's and literally do the book on the head thing, be drilled in posture, and indeed have a table meal where you learn formal table manners.