The verb “to cabbage” seems unrelated to Cockney Rhyming Slang:
The OED, for this meaning, suggests
Etymology: Probably < cabbage n.3 (see discussion at that entry).
In quot. 1703, the Spanish original uses hurtar, the usual Spanish word for ‘to steal’.
1. transitive and intransitive. Of a tailor or dressmaker: to appropriate (offcuts of cloth) as a perk when cutting out clothes. Now rare.
1703 P. Motteux et al. tr. M. de Cervantes Hist. Don Quixote III. ii. xlv. 438 He could not but imagine that..I had a mind to Cabbage some of his Cloath [Sp. él..debióse de imaginar..que..yo le quería hurtar alguna parte del paño].
However, this seems unlikely as the Spanish for "cabbage" is nothing like "hurtar" and "hurtar" sounds nothing like "cabbage"
Looking at the noun, which is recorded earlier:
 1. Offcuts of cloth appropriated by tailors and dressmakers as a perk when cutting out clothes.
1663 Hudibras: Second Pt. iii. 56 For as Taylors preserve their Cabbage, So Squires take care of Bag and Baggage.
1935 A. J. Pollock Underworld Speaks 17/1 Cabbages, bolts of stolen woolens.
1986 Sunday Express Mag. 6 Apr. 86/3 Either the cutter or the proprietor might sell the 'cabbage' privately and not put the money through the books.
Here, there is a strong possibility that "cabbage" is the vegetable because, when preparing a cabbage, the leafy part of the vegetable is stripped from the thicker white veins of the leaves of the plant. There, thus, appears to be a parallel in a tailor cutting out the cloth by following the white lines of chalk that the he uses to mark the cloth for cutting.
Added to this is the "-age" suffix can be seen as an ironic reference to the "cabbage" being some sort of "tax" on the cloth that is taken by the tailor:
- Forming nouns denoting a charge, tax, or duty levied on what is denoted by the first element, as ballastage n., housage n., poundage n.1, rowage n., etc.
This use of "-age" was very much standard pre-18th century.
In the entry for the noun, the OED has a long comment
However, The OED in the entry for "cabbage" noun = stolen off-cuts:
Compare the following quots.:
1648 R. Herrick Hesperides sig. G2v Thou who wilt not love, doe this; Learne of me what Woman is. Something made of thredand thrumme; A meere Botch of all and some. Pieces, patches, ropes of haire; In-laid Garbage ev'ry where.
1648 R. Herrick Hesperides sig. Aa3v Lupes for the outside of his suite has paide; But for his heart, he cannot have it made: The reason is, his credit cannot get The inward carbage for his cloathes as yet.
The first quot. clearly shows an instance of garbage n., probably in a contextual specific sense ‘shreds and patches used as padding’. The second quot. may show a variant of garbage n. (although if so, that variant appears to be unparalleled elsewhere), and it could therefore be taken as providing evidence for a transitional stage between garbage n. and the present word.
And yet, if there were a word "garbage" why would another be required? And why would anyone steal "garbage"?
Alternative etymologies are that the word is < French (now slang) cabasser to set (goods) aside, to steal, to cheat, deceive (second half of the 15th cent. in Middle French), extended use of cabasser to put (goods) into a basket (15th cent.; < cabas basket, panier: see caba n.), or that it is related to Middle French cabuser to deceive, cheat (1405; < ca- , prefix (see cabbage n.1) + abuser abuse v.) and its deverbal derivative cabuse imposture, trick (c1450). However, neither of these suggestions would account for the order of senses in English; both at the noun and at cabbage v.2 the earliest attested senses relate specifically to practices of tailors, rather than to cheating or stealing in general.
On the other hand, this seems reasonable.