If you are talking about Portugal and slavery, you are actually talking about Brazil.

I do not know the justifications that Portuguese people used to morally justify abolishing slavery in the metropolitan region of Portugal while letting it rage and profiting from it outside its main territory. Maybe someone more knowledgeable in Portuguese history can pitch in. I'll tackle the second part of your question.

The short answer to the second part of your question is the Aberdeen Act, followed by The Eusebio de Queirós Law. But let's talk about slavery in Brazil for a bit.

Slave trade and slave owning in Brazil was quite different from the US. While less than 10% of individuals kidnapped in Africa were brought to the US, above 40% were sent to Brazil (Thomas, The Slave Trade, The Story of the Atlantic Slave Trade). I don't have precise numbers, but it is safe to say that most of Portuguese slave trade was conducted to feed Brazil's hunger for more workforce and to replenish Brazilian farms where conditions were so harsh that the slave population could not reproduce in rates that would maintain their populations. For comparison, life expectancy in the American South in 1850 was 40 for white people and 38 for black people (Fogel, Time on the Cross: The Economics of American Slavery). In Brazil, white people had a life expectancy at 27 and slaves at 19 (Schwartz, Segredos Internos: engenhos e escravos na sociedade colonial (1550-1835)). Most of these deaths were infant mortality. We don't have good numbers on black and white child mortality in Brazil for this time, but we have some records for slaves showing values reaching 40% (comparing to 21% in the US). Emilia Viotti da Costa, probably one of the most important Brazilian historians of slavery and black history in South America, writes in the chapter O Escravo na Grande Lavoura from the monumental História Geral da Civilização Brasileira (vol 5) that slave infant mortality was 75% on "good" plantations and reaching 88% in "bad" plantations. The discrepancy between these numbers comes from the lack of records from infant deaths in slaves. This mortality, allied with more brutal conditions in Brazilian plantations comparing to the American counterparts, and you have an insatiable demand for African slaves starting when plantations in Rio became widespread (around 1750).

On a small tangent, the government fought this decline in slave population growth not by improving slave conditions, but by promising to free slave women that had more than 10 children. Since fertility is proportional to nutrition and inversely proportional to stress, this was almost impossible to achieve.

With so much demand, Brazil kept importing slaves well after its independence (1822) and its economy would run heavy on slave labor and agricultural exports, with coffee as the main product followed by sugar. This would change, however, with the Aberdeen Act in 1845, when England gave the the Royal Navy the authority to stop and search any Brazilian ship suspected of being a slave ship on the high seas, and to arrest slave traders caught on these ships. The result of the law was a surprising increase on slave imports, from 50k in 1846 to 60k in 1848, by slave-owners anticipating the end of the trade.

In 1850 Brazil, Argentina, Uruguay and Paraguay would begin the first of a series of conflicts in the region (The Platine Wars). Eager for international support, specially British naval support, and noting that slave trade was a dying practice, the Brazilian empire decided to abolish slave trade completely. This move was endorsed by the then Justice Minister Eusébio de Queirós and the law gets his name. Slave trade was abolished in Brazil and the search for the replacement of the slave had begun.

To counter the effects of the decline of the trade, instead of improving conditions or changing the economical structure of labor, the Brazilian empire slowly implemented immigration changes to "phase-out" the slave population, replacing it with Germans and Italians, whose countries were ravaged by war and unification in the second half of the XIX century. It is not by accident that Brazil has the largest concentration of Italians descendants outside Italy (one of them is writing this answer, my great-grandfather, born in 1903, was the first of my family to be born in Brazil). The conclusion is that slavery was finally abolished in 1888, black farmers were replaced by European migrants and were deprived of much of the country's growth. While slave trade dwindled after the Eusébio de Queirós law, slavery would still ravage the country for almost 40 more years and its consequences are still with us today.

Marques de Pombal abolished slavery in Portugal proper, in 1761. That law left slavery and the trade between Brazil and Guinea, Angola and Mozambique unaffected, because the government in Lisbon realized that slavery was fundamental to the plantation economy in Brazil. Similarly, the export of slaves from Angola was fundamental to that colony's economy.

The idea of slavery being abolished in the metropole (Portugal) while legal in the colonies was not unique. In the UK, the Somerset case in 1774 held that common law in England and Wales did not recognize chattel slavery, and no parliamentary law supported it. However, the Somerset decision did not speak to the legality of chattel slavery in Scotland (which had a separate legal system) or in the colonies where slavery was enshrined in law.

Could you provide the source you're using for these numbers?

I haven’t seen an answer to the last part of your question: why the sudden decline after 1850. A large part of the answer may have to do with shifting insurance practice and actuarial tables. Even clandestine slaving by Brazilian and Cuban concerns was often insured by legitimate insurance companies such as Lloyd’s. By the mid 19th century, it was no longer profitable to insure these voyages. As finance dried up, the trade began to decrease. A prosaic answer, and only one factor. Another, certainly, was the shift to ‘legitimate’ trade goods in coastal west Africa, particularly palm oil. The strategic necessity of labor intensive palm oil to both African kingdoms and British industry meant that there was a commensurate attempt to force the termination of the disruptive slave trade.