What is the origin of 'cash'?
Solution 1:
OED shows various citations for cash as "coin, money" between 1590 and 1600.
1593 G. Peele Famous Chron. King Edward the First sig. H, Now the Frier is out of cash fiue Nobles, God knowes how he shall come into cash againe.
1596 T. Nashe Haue with you to Saffron-Walden sig. O2v, He put his hand in his pocket but..not to pluck out anie cash.
1600 Shakespeare Henry V ii. i. 110 Nim. I shall haue my noble? Pist. In cash most truly paid.
The East India Company was granted a Royal Charter in 1600, but three ships had sailed to India in 1591, presumably returning within twelve or eighteen months.
OED's first citations do not indicate the word was in use with this meaning prior to the ships' return.
Solution 2:
I think OP's original instinct, that cash and caisse are cognate, was right. Casse or chasse (from L. capsa) appears to have been the usual French spelling (and, presumably, pronunciation) in the 14th C, and the word entered English at the same time. In both languages it had the general meaning “box, coffer”.
At the end of the 16th century it appears to have attained a European currency [sic!] in the additional narrower sense—derived, I suspect, from the merchant’s cash-box—of “cash-desk, till”. We find this in Florio's 1598 A Worlde of Wordes, Or Most Copious and exact Dictionarie in Italian and English:
And this in Randle Cotgrave's 1611 A Dictionarie of the French and English Tongues:
In these definitions I take it that counter has OED 1’s Obs. meaning 3. “A table or desk for counting money, keeping accounts, etc.”. (OED 4. is clearly derivative, but not attested until 1688 “A banker’s or money-changer’s table; also, the table in a shop on which the money paid by purchasers is counted out, and across which goods are delivered.”)
Cotgrave, moreover, not only recognizes caisse as an alternative French spelling, he provides a French parallel to the English usage (Prov. I suppose to mean “Proverb” rather than “Provençal”):
(By the way—cashier in this sense is derived from Fr. casser “break, annul”, and refers to breaking up a company of troops, not to paying them off.)
Occam's Razor suggests that there is no need to look to derivation of cash from an Oriental term which according to OED 1 is attested for this period only in the Port. spelling Caixa—which was itself a falsely-friendly application of the Portuguese cognate of casse/cassa < capsa.