Consider the terms artisan and artisanal. From en.wiktionary, artisan means “A skilled manual worker who uses tools and machinery in a particular craft”, while artisanal has senses including “Of or pertaining to artisans or the work of artisans” and “Involving skilled work, with comparatively little reliance on machinery”.

As a parallel to the craftsman/craftsmanship or craftsperson/craftspersonship pairs mentioned in the question and in some answers, we have artisan/artisanship.

From en.wiktionary, artisanship means “The state or quality associated with being an artisan”; thus, it denotes working in a skilled manner.

Oxforddictionaries.com gives the following definition and two examples for artisanship:

Skill in a particular craft:

‘pieces of jewelry which testify to the high artisanship of these ancient people’

‘a heritage of exquisite artisanship’

Note: a pleasing and workable gender-neutral alternative to “sportsmanship” seems less available. Interestingly, Google ngrams for sportsmanship, sportswomanship, sportspersonship (or, more clearly, for sportswomanship,sportspersonship) shows that sportspersonship is used far more frequently than is sportswomanship.