The two concepts aren't really that opposed, the core concept of universal suffrage is "all adult citizens can vote" not "every single person can vote". No country that practices what might be called "universal suffrage" today comes close to letting everyone vote; some portion of non-citizens (in some cases even if long-time residents), those under a certain age, the severely ******** or insane, criminals (especially while jailed), and other exluded categories aren't allowed to vote. It would be very easy to have a system where slaves aren't considered citizens (like US pre CW), or suffer some other disqualification.
Even if we presume that slaves can vote, it still doesn't mean that they can vote themselves free. Laws or traditions can make it difficult or impossible for a 'free the slaves' candidate to come to office; for example, in the US today there are no credible candidates that I could vote for to end the War on Drugs, and it would be easy to have a country with a consitution (or equivalent) that specifically allows slavery that also requires that all candidates swear to uphold it. The votes of slaves can be controlled more easily than those of industrial workers (since slaves can't just move away), so a vote collecting setup like Taminy Hall could work even more easily to have slaves come in and vote with one of their owner's men double-checking the ballots.