Was there a "literate class" outside the Shi in Zhou-dynasty China?

Officially, no -- there was not another Scholar (shi [士]).

Culturally, the literati as a social class did not exist during a large part of the Zhou dynasty, and only came into existence in late Zhou (Warring States) period.

During this period of late Zhou (I am narrowing this to Zhou because that was the question), the populace was segregated into four occupations (roles), in descending order of rank:

• Gentry/Scholars, shi (士) -- the elites, i.e. military commanders -- this class is the literati we've been looking at.
• Agricultural Producers -- nong (农) -- peasants, producers of foodstuff.
• Labourers -- gong (工) -- craftsmen. artisans, etc.
• Merchants -- shang (商) -- lowest because they do not produce (grow or create) their own.

This is the 'broader societal class' part of the question.

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But then it goes to mention merchants (as one example), and asked if there were trained as scholars. The short answer is they will not be trained per se by the Imperial court (based on the classification) but they can of course learn on his/her own. So, no scholarly training for merchants and slaves.

Finally, if you're asking if the title of Scholar (shi) is restricted to this hereditary class. Yes, it was, and the others (farmers, artisans & merchants) could not become Scholars via the Imperial (Civil Service) Examination, which was only reserved for descendants of the Shi. Only much later, 6th century CE, could others try to qualify as a scholar and find work as a court servant (bureaucrat), i.e. during Sui Dynasty.

Having provided this answer, and because we are of course narrating based on available textual material (and some material culture), it does not mean that ancient China is exactly as explained -- see below for context.

More Context

There's a problem with providing historical answers that are not sufficient, i.e. missing details. The natural temptation to be more thorough is, however, countered with a need for expediency; such is the need for convenience in our time-starved society today.

So, I'd like to ask readers of this answer to keep in mind the following context (as a frame of reference, which might help understand the narrative of the answer):

• Zhou dynasty (c.1050-256 BCE) - can be viewed as earlier Western Zhou (c.1050-771 BCE) and later Eastern Zhou (770-221 BCE)
• Eastern Zhou should be separated into sub-periods of Spring and Autumn Period (770-c.480 BCE) and Warring States Period (c.480-221 BCE).
• Western Zhou is the idealised unified elite culture of many Chinese classical texts such as Confucius (c. 551-479 BCE) Analects, Mencius (371-289 BCE) Mencius, Xun Zi (Hsun Tzu) (298-238) Xunzi -- none of the classical Confucian historians/theorists lived in Western Zhou era.
• The Later (Eastern) Zhou was a period of decline and much confusion, i.e. warring states, 100 Schools of Thought --- all competing for legitimacy with the Imperial court(s).
• Into this mix, we now have from Qin (at the latest) and late Zhou (at the earliest), a form of social control via the Four Occupations classification. In terms of how well it reflected the society of ancient China, the best we can say is that it is based on an idealised structure.

I do not have the citations on hand, but I believe Feng Youlan and Edward Shaughnessy have proposed that the Mohist movement may have taught writing to students outside the Shi class. IEP says:

Some scholars speculate that Mozi and the Mohists probably came from a lower social class than, for instance, the Confucians, but the evidence is inconclusive and at best suggestive. Nevertheless, if the conjecture is true, it could well explain the often repetitive and artless style in which much of the Mozi is composed and the anti-aristocratic stance of much Mohist doctrine, as well as why the Mohists paid such attention to the basic economic livelihood of the common people.

IEP is being polite. The book of Mozi is famous for its awful writing style (frequently amusing in English translation), and many people have suggested that it was written by multiple writers who lacked education in rhetoric and style, suggesting entire schools of pragmatic literacy outside the traditional literati.

The answer has to be no. In China, there were the four basic occupations at the time: the shih (literati), nung (peasantry), kung (artisans) and shang (merchants). Each occupation had its job and was supposed to focus on it, and the job of the shi was to fight, and later, to administrate, and to engage in scholarship. Thus, speaking of a literate class, the shi is the only one.

In Zhou dynasty China, membership of the shih class was mainly hereditary: to become a member of this literati class, one needs to be born into it. Although Confucius himself promoted education without barriers, i.e. yu chiao wu lei 有教無類, and actually did take students of humble descent (turning them into shih), the vast majority of shih acquired their status by birth.

Of course, when the k'o-chiu 科舉 examinations were introduced later, membership of the shi class became based on meritocracy, namely passing the exams.