Why was it tradition to offer 'half-baked cake' to departing students?

TL;DR: I doubt that there was such a tradition.

It would be nice to see more of the context from Punch’s Snapdragons for Christmas, but even given the terse reference in Hearn, we ought to take into account that Punch was a humorous and satirical magazine, so if it said that it was “customary” at English boarding schools for the masters to offer “half-baked cake and home-made wine” to their departing charges, then the intention may have been satirical or ironic.

If so, my interpretation is that the author was satirizing the penny-pinching of the boarding schools: that is, the allegation was that the schools were skimping on cooking fuel in order to screw more profit out of their charges, without regard for their health and welfare. The cake was “half-baked” because it was taken out of the oven too soon in order to save on firewood. Compare the following passage from the memoirs of John Howard, which is making a similar allegation but without the irony:

He [John Howard] visited also in this tour three out of the four nurseries for the reception of children, from two to six years of age, and he paid the more minute attention to their condition, because their tender years rendered them incapable of struggling with hardships, or of making complaints. He was sorry, therefore, to find the same gross neglect of their health and cleanliness, as disgraced the schools for the elder children. The master of that at Monastereven pretended to be an apothecary, but a pretty correct judgment may be formed of his medical skill, from his giving all his infant scholars regularly sulphur and milk for their breakfast, and from his declaring his intention of having a general anointing for the itch, whether they had, or only might have, that unpleasant disorder, though their beds and their persons were certainly quite dirty enough to give it to them all round. It was still further proclaimed too, by the uncommon mortality amongst his nursling patients, for whom in one quarter’s bill, there was a charge for eleven coffins. At the time Mr. Howard visited these most pitiable objects, they were dining, at three o’clock, on potatoes not properly boiled; five or six of the most sickly being indulged with a piece of half-baked cake or bread, but drinking the common beverage of the whole, sour butter-milk.

_{James Baldwin Brown (1818). Memoirs of the Public and Private Life of John Howard, the Philanthropist, p. 527. London: Rest Fenner.}

The passage in Dickens has a similar meaning: the master of Scrooge’s old school was just as miserly as those of the schools visited by Howard.

It wasn't — “curiously light wine, and a block of curiously heavy cake” was intended by Dickens to convey that the guests were selfishly given stale Christmas cake and watered-down wine. According to Victorian Web's commentary on A Christmas Carol:

Brock demonstrates that Scrooge's sole sibling, gleeful little Fan, is smaller and younger than her brother as her feet do not quite touch the floor as the head-master entertains the pair in his "shivering best-parlour" (35) with the "curiously light" (watered-down) wine and "a block" of stale cake.

Which very much matches Arthur Rackham's 1915 illustration for the book of the children staring apprehensively at their "dainties".

After searching the internet and British newspaper archives extensively, the only reference comes from A Christmas in Kent from The People's Journal where the phrase also appears to be intended for humorous effect.

"Fine seasonable morning!" said my uncle, brushing off a few flakes of erratic snow from my cloak, as we drove into the yard of the Red Lion. "Something like Christmas weather, this; it reminds one of going home for the holidays;" and the old gentleman rubbed his hands gleefully at the recollection. "It makes one feel almost a boy again," he continued, laying his arm warily on that of the ostler, and feeling very carefully with his stick before he trusted to the footing-"Just such mornings-bless me, how well one remembers them!-up before daylight-bolstering one another till breakfast time-too happy to eat-turning the schoolroom out of the window till the half-baked cake and home-made wine were handed round; then the coach, and our hasty leave-takings...