I can't say that no one ever worried about a Yankee VP doing a tie-breaker, but I've never heard of it and there's basically no room for such a thing to happen in the political system of the time. Part of this is that the two-by-two (slave & free states admitted in rough conjunction) balancing act and the importance of keeping the Senate evenly split is a pattern that emerges after the war of 1812. For many white Americans in the North, until then it wasn't clear that the advance of freedom stalled at the Ohio River and the Mason-Dixon Line. That there is a clear sectional cleavage over slavery qua slavery, which may override other partisan concerns, doesn't become apparent until the Missouri Crisis. The era we're looking at, then, is roughly 1820-1860.

In that period, this would only matter there was a Northern president who wouldn't veto any law passed by a tie-breaking vote. That's both extremely unlikely in its own right as both parties have constituencies in both sections and so are adverse to making an attack on slavery into a party measure. They're actually much more prone, along with many other northern pols, to vote for proslavery measures. Many, including extremely prominent senate leaders as well as presidents and Cabinet officials, are so pliable for slavery that the earn the name doughface. This is politically possible for them because the white South, at least by 1800, is overwhelmingly in favor of keeping slavery in perpetuity and prepared to make that the sine qua non of their politics. Northern whites do not feel this way as frequently and intensely until about 1860.

Furthermore, parties balance the ticket sectionally. A Southern president will probably have a Northern VP. A Northern president always has a Southern VP. (This also extends to cabinet offices, though balancing there is more fine-grain regional.) Even counting very inclusively, Yankees in the nation's highest office between 1820 and 1860 are John Quincy Adams, Martin Van Buren, William Henry Harrison (blink and you'll miss him), Millard Fillmore, Franklin Pierce, and James Buchanan. Their VPs run like this: SC (Calhoun, no less!), KY, VA (proslavery far-rightist John Tyler), no VP (Fillmore became president when Taylor died), Franklin Pierce (AL, then nobody because he dies on April 18, 1853), KY (runs on the Southern Dem ticket in '60, becomes Confederate general).

So on the one hand you have a Southern president who will surely veto any antislavery legislation that squeaks by. On the other, you have a Northern president with a southern VP who would probably not cast a tie breaking vote on an antislavery measure, except to vote against it. The political norms as they exist in the Second Party system prevent such a thing, essentially by design.

However, it's unlikely that an actual, meaningful proslavery measure would have an easy time getting to half the Senate for the same reason that a Northern president before Lincoln is unlikely to let one become law: the doughface problem. To win the presidency, one needs to be amenable to doing the bidding of slavery. To be successful in the Senate, you need to function in a party that has a southern wing. There's more room here to be antislavery, since you're not directly answerable to southern voters, but there are practical limits still. In a fairly strongly antislavery state one has room to go further but most Senators see a president in the mirror every morning, or at least a Cabinet secretary. Going out on antislavery limbs is a great way to make sure that never happens. With the exception immediate lofty ambitions, similar dynamics tend to play out among Northerners in the House. But since the sectional majority is big and the constituencies smaller there's considerably more political space to strike out aggressively against slavery.

These norms begin to crack up during the Mexican War, but they're still powerful enough -especially when joined with personal ideological commitments- to bind career proslavery Yankees like Pierce and Buchanan all the way to 1860. Van Buren isn't going to push against them while president because he knitted back together the old Jeffersonian proslavery coalition after the Missouri crisis. Fillmore has a less conspicuous proslavery record before the presidency, but he hailed from the further right faction of New York Whiggery and delivered big time for slavery by signing the Compromise of 1850.

Good question...in theory the VP was expected to carry out the policy agenda of the POTUS' party, and therefore the VP's potential actions weren't specifically an issue beyond the general presidential partisan issues.

However a more germain question would be, what if POTUS died and the VP became POTUS? Would new President have policies consistent with his predecessor?

As it turned out, two Presidents died in office in the antebellum years (both of the Whig party). And in both cases, the newly minted POTUS departed dramatically from what the old POTUS likely would have done.

Tyler was rabidly pro-South and pro-slavery, unlike the more moderate (although not abolitionist by any means) Harrison. Tyler's crowning achievement - accomplished at the last minute while he was a lame duck - was to engineer via legislative means the annexation of Texas specifically to extend the slave states further west. This set the stage for war with Mexico and the opportunity to create even more slave states in the west - which Polk gratefully pursued with extreme zeal.

Fillmore was a short-sighted politician with primarily regional New York State interests. He was openly racist and xenophobic, participating in the "Know-Nothing" and Anti-Mason movements. Taylor had been a slave owner (like many of the Whig leadership) but he was not keen to see slavery expanded to the far west, and as a military man wasn't adverse to protecting the Federal objectives in the west at gunpoint. Fillmore however allowed himself to be cowed by southern factions into approving the notorious Compromise of 1850, which negated the Missouri Compromise, permited slavery in theory to expand north of the Madon-Dixon line, as well as creating the Fugitive Slave Act, which would have forced Northerners to participate in the capture of escaped slaves. The Fugitive Slave act created such rancor in the north that essentially the Whig party was destroyed, opening the way in a few years for the advent of the Republican Party, the election of Lincoln, and the secession of the sothern states.