The Oxford dictionary of word orgins supplies:
chuck L16th
This informal word meaning ‘throw’ is the same as the one meaning ‘touch (someone) playfully under the chin’, probably from Old French chuquer, ‘to knock, bump’ (of unknown ultimate origin). The chuck [L17th] of a drill is a variant of chock, with chunk [L17th] another variant. The phrase the chuck expressing rejection (give somebody the chuck) dates from the late 19th century, while the sense ‘to vomit’ is an Australianism from the mid 20th century.
The OED supplies one usage from L16th:
1593 Prodigal Son iv. 112 Yes, this old one will I give you (Chucks him old hose and doublet).
This example cited in many books, including Slang and its analogues past and present.
The text of the play is found in: The school of Shakspeare (1878).
The Concise Oxford Dictionary of Etymology says:
Chuck
give a playful touch under the chin; throw with the hand XVI.
Also (dial.) chock XVI. perh. — OF. chuquer, earlier form of choquer knock, bump, of unkn. orig.
Wheat's An Etymological Dictionary of the English Language (1898) has an entry for chuck and states that the word is a doublet of 'shock', which also comes from Old French choquer. But I didn't find any uses in the OED of 'shock' to mean 'chuck/throw'.
Klein's A Comprehensive Etymological Dictionary of the English Language (1971), p 286 notes:
chuck
tr verb, to throw--From earlier chock, from French choquer, 'to shock', which is probably borrowed from Dutch schokken, a word of imitative origin. Compare shock, 'to collide'.
and
An Etymological Dictionary of Modern English (1921) says:
chuck. To throw. Earlier chock. F. choquer, of doubtful origin. Earliest E. sense is connected with chin.
Best Answer
'Moke' in U.S. slang
As a U.S. slang term, moke has a very problematic history. Though its earliest slang meaning (going back at least to 1839), was "donkey or mule," it was also used for decades as a disparaging term for "a black or dark-skinned person" (starting not later than 1847) and "a foolish or inconsequential fellow" (starting not later than 1855).
J.E. Lighter, Random House Historical Dictionary of American Slang (1997) identifies three basic meanings of the term in connection with living creatures:
So historically in U.S. usage, a moke could refer to a beast of burden; a black or dark-skinned person; or a fool or lightweight.
George Matsell, Vocabulum; or The Rogue's Dictionary (1859) is one of the earliest glossaries to list moke:
James Maitland, The American Slang Dictionary (1891) indicates that the first two meanings cited by Lighter were used in different parts of the world:
Theories of the term's origin
As for the source of the word (which Lighter says is unknown), different theories have appeared over the years. Maximilian Schele de Vere, Americanisms: The English of the New World (1872) has this for moke:
"Billy Bilger" [Andrew Pendleton], The Silly Syclopedia: Containing "daffynishuns" of the Words of the Slang Language Spoken by the Midshipmen of the U.S. Naval Academy (1908) has this:
An illustration of "mess hall moke" in the same book clarifies that Naval Academy midshipmen used the term to refer to (African American) food servers and table bussers in the mess hall.
Harold Wentworth & Stuart Flexner, Dictionary of American English, first edition (1960) has this:
To sum up, the suggested source words are murky (suggested in 1872), mozo (suggested in 1908), and mocha (suggested in 1960).
The Romani connection
John Hotten, The Slang Dictionary: Or, The Vulgar Words, Street Phrases, and "Fast" Expressions of High and Low Society (1864) alludes to an alternative etymological explanation that may be of special interest to the poster:
The 1874 edition of Hotten, The Slang Dictionary: Etymological, Historical, and Anecdotal, expands this entry considerably:
Hotten's dictionary was published in London, and he is clearly concerned with the British English meaning of the slang term moke. A letter by William Cumming Wilde, published in American Notes and Queries (December 22, 1888) follows up on the etymological question:
These references suggest sources of moke in Arabic al mocreve (in 1874), in "English Gipsy" mokar (in 1875), in Swedish mocka (in 1877), and in Old English mokey (in 1888).
Conclusions
Whatever the original source of moke in the British English sense of "donkey" may have been, the fact that it was identified with "English Gipsy" slang by 1864 and that it appears to have been used as a term of endearment (or perhaps of affectionate exasperation) by a Romani grandmother more than a hundred years later is fascinating to me. To judge from the OP's anecdotal evidence, the Romani English sense of moke does not appear to have been tainted by the derogatory U.S. English sense of moke as "dark-skinned person."