The Desires
The standard dictionary definition of craven is "cowardly":
This could be illustrated through Ancient History in the conduct of Ephialtes, the "Judas Iscariot" of Greece, who in satisfying his craven desires, brought untold misery to his fellow patriots by his treachery... Henry Germanus Maeder, Educational Outcomes Unique to Ancient History, 1927
Ephialtes betrayed the united Greek forces to the Persians at Thermopylæ in hope of gain, his craven, i.e. cowardly desires. Greed in itself is not always cowardly, but accepting payment for betrayal, as Judas Iscariot, may certainly be deemed so.
In other usages, however, moral failings other than cowardice seem to be on the author's mind: craven desires are base, often debauched, and in the fiery rhetoric of Christian moralists, a grievous sin. For those wanting to perfume their otherwise wanting prose with a whiff of scandal, the word may equally serve.
They laugh and dance, indulge and spend, and find some satisfaction in pursuing thrill, excitement, danger, popular acclaim, and material security. They follow their selfish instincts or yield to their craven desires. Leslie F. Brandt, Jesus/Now, 1978.
Your religious training may have taught you that the body is a repository of base instincts and desires that must be repressed, ignored and squelched, lest you accede to your craven desires and spend eternity regretting the decision! Ramón Stevens, Whatever Happened to Divine Grace, 1988.
And now for someone apparently ready to chuck it all for (shudder!) one night of pleasure:
She entered the living room in a most charming negligee. She had never interested me sexually in all the years I had known her — somehow I had never thought of her in that way. She had always been on a pedestal above all such craven desires. Her seductive negligee plus the liquor was sufficient for me to discover charms never before unfolded. William Ward Smith, A Letter from my Father, 1976.
This isn't cowardice; it's sex, drugs and rock 'n' roll. How lexicographers have missed this usage is beyond me, but only Superman-sized conceptual leaps can get to cowardice from charming, seductive negligees.
Craven Desire in the Singular
Craven desires can just as easily be condemned in the singular, where the distance from the original meaning of "cowering, cowardly" is even more apparent. Occasionally, the familiar moral tone of the plural is still heard. Thus paraphrasing the Buddha:
the craven desire is the root of all suffering
And a Christian writer in an imaginary conversation with Hieronymus Bosch:
...the craven desire is for immediate gratification. It uses the other person, or thing, or even God, to meet this consummation of desire.
This suggests a meaning of inordinate, selfish desire, or, in a word, lust, which can be described negatively in countless scenarios without audible religious overtones. Thus a craven desire
for power, again here, for media attention, attention and power, money, to stay in power, for publicity, to be liked, to seem young, purely for commercial purposes (Hannah Montana), to be provocative (Rolling Stones), or to be on television (reality stars).
One writer turns to Norman Mailer for a critique of the media in the run-up to the Second Gulf War:
In an effort to analyze our wanton and exuberant Iraq war-lust, Mailer points to our craven desire for manipulated and televised displays of dominance as a major factor for the drive to the military invasion of Baghdad.
A journalist takes a look at the 1936 Berlin Olympics and
the Germans' craven desire to paint a sanitized picture of their society to a skeptical world.
While craven is never completely free of opprobrium, there are cases where the desire is neither overwhelming nor morally suspect, merely very strong. Thus a business has a craven desire to win over consumers or fill a niche market, and the director of the film Benjamin Button has a craven desire to elicit emotion.
One writer suggests that US senators' reluctance to use email comes from their craven desire to avoid record-keeping, while the Far Right in the US has a craven desire for academic acceptance.
Finally, a self-designated perfectionist has a "craven desire to do everything perfectly the first time" and a woman in India describes humorously her craven desire for a Michael Kors handbag.
No stretch of the imagination can inject cowardice into any of these usages of craven.
The Idols
Craven idol received no hits in a Google NGram; a collocation search only yielded results such as fear/fears, unambiguously in the sense of 'cowardly.' Since Craven Idol is also the name of a heavy metal band, googling becomes a frustrating exercise in using the minus sign. What non-metalic results appear, however, suggest that the band is not using a nonce word for its name.
A Time Magazine review (31 Mar 1961) of a decidedly anti-modernist book by Robert Elliot Fitch, Dean of the Pacific School of Religion, is entitled The Craven Idol.
The Golden Calf of the Exodus saga is craven:
But there in that wilderness/
before the craven idol/
before the king demanding to be god/
Refusing to bow, to deny the creator/
honoring the words written so long ago/
Prepared to be burned, to submit to the furnace/
entering in, faithful to God. Raymond A. Foss, The First Commandment, 2009.
God was about to destroy the children of Israel because of their craven idol that [they] had made and also because of Aaron's sin. Erica Michael, Birthing His Glory Through Dance, Part 1, 2010.
Foss also alludes to the statue of the Roman emperor to which Christians were to pay homage or suffer punishment.
In The Invisible Dragon: Essays on Beauty, Revised and Expanded, 2012, author Dave Hickey describes a painting of Jesus, which
... thanks to oil glazing, seems an uncanny incarnation of Christ? Is it a picture, an icon, a craven idol, or something else?
A four-item catalogue such as this is the last place one would expect an affect-laden or polemic adjective, though idol is the only noun that gets any adjective at all. Whatever the intentions of the author, pausing in the middle of a list to comment on the "cowardly" nature of a cultic statue is not likely to be among them.
Almost all of these uses of craven, whether collocated with desires or idol, come from literature with an overtly religious stance, or certainly a moral one.
For the prophet Habakkuk (2.18), graven or molten images are made from human hands: they are "dumb," inert stone or metal incapable of speech. In 1 Cor. 12.2, they are merely "dumb idols." It would be difficult to imagine a writer of devotional literature ascribing any affect at all to an idol, even a negative one such as cowardice.
The Dilemma
Craven desires and craven idol seem to inhabit the same Christian moral and polemic universe, yet the usual definition of craven as "cowardly" is a poor fit for either. A meaning of "depraved, debauched" might work for desires, but not for idols. Try as I might, I can only see the use of craven for idols as a strange variant of graven, even by those intimate with the latter term from the Hebrew and Greek Scriptures.
The parallel of bronze peacocks and craven idols in the Ottoman history would suggest an adjective equally parallel to bronze, and none of the sources for craven idol would suffer in the least were graven used in its stead.
The use of the singular diverges just as strongly from dictionary entries, yet it's clear that the word craven is being used in ways other than that prescribed. Perhaps craven suggests to some speakers an excessive craving for most any desired object, which can be acknowledged but not necessarily condemned. The disparity between living language and lexicography for this word, however, remains an intriguing dilemma.
Best Answer
I see nothing inherently wrong with employing source as a verb in either your sense or the commercial sense which Google afforded you.
It earned this sense by exactly the same process which gave rise to the established academic term to document, a similarly verbed noun which similarly is open to two constructions: to provide references to existing corroborative sources or to create a record which will constitute future corroborative sources.
Your only concern should be what audience you are addressing—some will consider your use of source breezily fresh, others cheesily faddish—and whether you care how they take it.
For the time being, source is slang and document is jargon. You pays your money and you takes your choice.