Not even "absolute" monarchies were absolute.

Quite obviously, a monarch cannot rule the kingdom directly. He or she depends on a vast number of henchmen and the belief of the population that the government should govern. Even if there is widespread dissatisfaction with the government, the kingdom will go on if bureaucrats, the nobles (if any) and the peasants all believe that rebellion is bad for them, personally, on the short term. The king's tax collectors could not possibly subdue a widespread peasant rebellion. They can probably get the first person to stand up and become a ringleader, and the second, and the third ...

The flip side of this coin is that the "absolute" monarchy will be toppled if enough people, in the right positions, say "No!" to the monarch. The royal dynasty will be aware of that and so will the merchants and nobles. The merchants should have a much better understanding of how the kingdom really works than the average peasant, after all.

The monarchs know just how far their subjects can be pushed, and how fast. They run a multi-generational campaign to discourage slavery, but they cannot ban it yet. Every couple of years, there might be a new law to better the position of slaves.

You could copy the frame of the Early Roman Empire and make the slaves prisoners of war or criminals sentenced to slavery.

Those slaves work in the mines or work the huge fields wealthy nobles have and that provide food for the society.

The Monarchs don't want slaves, because they can be dangerous. But still, the kings can't outlaw slavery because someone has to work the fields and do other dangerous jobs and there will be always prisoners or criminals. Slavery takes the burden or keeping them alive from the government and places it in private hands.

1. 
   

   "Why a royal family couldn't just outlaw the process altogether within its kingdom's borders, thus freeing the slaves?" Because slaves are expensive, and no king is rich enough to compensate the owners, not foolish enough to try to deprive their owners of their property.

   
   
   

   As they say, the quickest and surest way of losing one's authority is to give orders why one knows won't be followed. So slaves are both expensive and numerous; thus setting them free without compensation would induce massive economic loss; so who's going to pay for their freedom? Is the king rich enough?

   
   



2. How many are those slaves, really? In the real European Middle Ages slavery was legal, slaves did exist, but they were mostly luxury items, not really numerous enough to represent a significant social problem. As long as the rich people in the kingdom don't have so many slaves that they could use them as private armies, what's the problem? The king is there to rule the country into peace and prosperity; antagonizing all the rich people in the kingdom is not conducive to peace and prosperity.




3. You say "chattel" slavery. This term is usually associated with the modern American society in the 18th and 19th centuries -- most other legal systems don't even have a concept directly corresponding to the Anglo-American "chattels". (The closest you can come in European legal systems is "mobile goods".) Please do not think of the American South circa 1850 as in any way representative of historical slave-holding societies; it was an historical aberration, with the institution of slavery so alien to the rest of the society that it caused a massive civil war.




4. Salaried work was quite uncommon in the real Middle Ages, and, where available, was not always a desirable station in life; by far the most numerous class of salaried employees were servants, a.k.a. "domestic help" if we are to used anachronistic terminology. How much is a servant paid in this society? Is there enough money in the economy to convert all slaves to salaried employees? Remember that when set free those people will be destitute -- they have no land, no position: how are they not going to transform from slaves into vagrants?

The characteristic of slavery IMO is that a slave can't (isn't free to) choose their master and their work conditions.

Perhaps the kingdom has some jobs that no free person would choose to accept -- work in a radioactive mine, for example; or, work in a brothel; or, work for "elves [who can be] more cruel than any human slaveowner" -- so that the only way to staff this work is with slaves.

Conversely of course the royalty have no problem finding people willing (even eager) to work as salaried employees.

Other possibilities -- the prevailing religion says that "might is right" rather than "blessed are the brotherly"; there's racism, nationalism, classism, etc., which let people see slaves as sub-human or non-human; the monarchs aren't "absolute" and there's no political movement/will to emancipate slaves.

Because the Monarch would face a rebellion

Lets do a thought experiment.

There was a movement to make George Washington King of the United States. Lets just say that happens. He rules over a land divided into a pro- and anti-slavery fashion.

Now, King George I of America dies. Who will be heir? George has no children, he had only a nephew (who died with no children) and an eighteen year old step-grandson. Not much to hinge a dynastic claim on.

Lets say the Continental Assembly goes back to the well and gets a new king, John Adams, who was the natural successor. Now John I did have a son, John II, or John Quincy Adams. John II would have been king from the 1820s to the 1840s. John Quincy Adams is also remembered as a fierce abolitionist.

What do you suppose would have happened if King John II had tried to abolish slavery? Or perhaps his son, Charles Francis Adams, King Charles I, if he had tried to abolish slavery in, say, 1861?

Pecunia non olet

Money doesn't smell, and when faced with economic reasons, even the most human royal family has to yield.

Slaves are a key part of the economic tissue, allowing the prosperity of the kingdom, ensuring cheap work force, flexible job market and cannon fodder when needed. And the royal family knows pretty well that prosperous and well fed subjects are way more loyal than starving and pocket empty ones.

Thus, while they can set the example by not directly using slaves, they secretly rely on slavery to secure their sit.

A History of Slavery

In the old days, slavery was actually the norm in your kingdom. It was in fact the entire basis of your kingdom's economy; a huge chunk of the market was the buying and selling of slaves, and the other sectors made heavy use of them. In other words slaves were literally the lifeblood of the system. It's likely that the royal family of the past took a much less negative view of slavery, but in more recent times they changed for the better(possibly because another dynasty took the throne).

Even though the newer royals want to outlaw slavery, they can't just do it right away. That would invite the kingdom's total collapse, a civil war and possibly a hostile takeover by a not-so-friendly neighbour. They have to slowly wean the nation as a whole away from slavery while gradually reducing the ground within which slavers have to operate, so that's what they've been doing over several generations.

In the meantime, while slavery is not totally abolished and slave owners/traders still exist, folks of all races who are(or think themselves to be)truly destitute would be "recruited" or throw themselves at the slavers in hopes of having food and lodging provided for them. This would contribute to the negative view most people would have of slavers; they'd be seen as bottom feeders who prey on the most vulnerable.

There's an excellent scene from Tamora Pierce's Squire which explains exactly why practices that the royal family abhors might still be around.

"It's not right," [the king] told Kel, to her profound shock. "Only a fool would say that it was. I am called many things," he admitted with a crooked smile, "but 'fool' isn't one. What do you want?"

...

"Change the law, sire."

"Change the law," the king repeated. "Squire, what do you think her majesty and I have done ever since we took the thrones? No, don't answer - I dread to think what you might have the courage to say. We have been trying to change laws - not this particular one, but many like it." He smiled bitterly. "The problem is that monarchs who wish to live until their grandchildren are born do not hand down any law they like. We must treat with our nobles, who are equipped to go to war against us; we must compromise with them. We must treat and compromise with merchants, who give loans for pet projects such as dredging Port Legann's harbor. We compromise and treat with farmers, who feed us, and street people, who can burn a city down. There are priests and priestesses, who can tell people the gods have turned their faces from the Crown, so they need not obey us. And the mages - I'll leave it to your imagination what mages will do when angered. Any law Thayet and I propose offends someone. We must balance opposing forces. Our successes vary."

Kings generally have more power than anyone they rule over. But they do not have more power than everyone they rule over. There's a limit to how heavily they can rule from above.

From actual history, the Magna Carta was signed after a group of barons got tired of the monarch's rule through force.