Spell check may have no problem with acquaintanceship, but the word in superfluous. The older and shorter (and easier to spell) acquaintance covers all acquaintanceship’s conceivable territory. The suffix -ship adds nothing.
Acquaintanceship could be shortened to acquaintance in each of these examples:
Sanford said the most significant influence on his life was his acquaintanceship with men in the factory. [Schenectady Gazette]
No marriage is required to accomplish this, not even an acquaintanceship or personal meeting. [Nigerian Observer]
Depending on their level of acquaintanceship with the army, some may pay lower rent. [Newsline]
Two men who met in a drug treatment program continue their acquaintanceship in jail, authorities say. [WSOC Charlotte]