It's actually not entirely clear why specifically the Satsuma leadership decided to invade.

Let me tell you what we are sure of.

Satsuma served as the central authority's way of contacting the Ryukyu Kingdom.

Whenever Hideyoshi's men, or during the time period of the invasion, Tokugawa's men wanted to send a message to the Ryukyu islands, they would do so by first sending it to Satsuma, where it would be entrusted to Satsuma messengers and relayed from there to the Ryukyu leadership.

Satsuma treated its relationship with the Ryukyu Kingdom with a fair amount of gravitas and import

Satsuma maintained diplomatic and trade relations with the Ryukyu Kingdoms for a long time and saw it as a companionable relationship, if a tad patronizing. It certainly took it upon itself to occasionally tell the Ryukyu Kingdoms what they thought the Ryukyu people should do along with missives from the shogunate.

Satsuma also clearly enjoyed it's preferential treatment for being the shogunate's main contact for dialogue with the Ryukyu Kingdoms.

It should be noted that the rulers of Satsuma, the Shimazu, were one of the 'disloyal' clans that sided against Tokugawa. But after the Battle of Sekigahara, they were 'forgiven' and while not given preferential treatment, were allowed to keep their status and land, remaining one of the strongest, richest provinces at the time.

The Ryukyu Kingdom would quite often slight the Satsuma lords and the Shogunate

The Ryukyu leadership would often ignore calls for tribute, war requisitions, and diplomatic niceties that Satsuma and the shogunate asked for. For example, during the invasion of Korea, the shogunate asked for soldiers and Satsuma suggested at least sending supplies and foodstuffs. The Ryukyu sent nothing. Other times, messages would be completely unanswered, which was considered rude to say the least and outrageous at worst.

**The Ryukyu Kingdom enjoyed a very lucrative and important part in maritime trade **

The Ryukyu Kingdom played a pretty huge role in maritime trade. It conducted major trade with China, Korea, Japan, Siam, Sumatra Vietnam, and various other kingdoms/major port cities in South East Asia.

Particularly the Malaysian kingdoms were a source of hugely lucrative trade, seeing how Malaysian spices (including pepper, cloves, and nutmeg) fueled a huge portion of the historic spice trade, which had Arabic, Chinese, and European buyers more than ready to pay exorbitant prices from any middle men (hint hint Ryukyu traders) knowing full well they would be sold for many times whatever they paid for them.

Satsuma's casus belli was to put the Ryukyu Kingdom in their place

The Ryukyu Kingdom's last rude gestures were, again, to ignore shogunate messages.

After Tokugawa Ieyasu's rise to dominance over all of Japan, Ryukyu's king was asked to formally recognize Ieyasu as the new shogunate of Japan, which was ignored.

Ryukyu sailors were also occasionally shipwrecked and were returned at the expense of Japanese lords. Despite requests to formally thank and show proper gratitude, Ryukyu again deigned not to respond.

In 1606, Satsuma asked permission from the Tokugawa shogunate to invade and it was granted.

What did Ieyasu Tokugawa have to say about the invasion?

After the invasion was complete, Tokugawa Ieyasu praised the Satsuma lord for a job well done. Presumably, even though he was technically 'retired', Ieyasu also gave the go ahead for the invasion in the first place.

Since this was before Sakoku, did this have an effect on Japan's trade relations with China, Holland, Spain or Portugal?

The biggest effect on trade relations was that trade with Ryukyu was one of the ways that Japan obtained Chinese silks without having to deal directly with Chinese trade missions. Satsuma control of Ryukyu maritime trade was a clear benefit to the invasion. But they also did not want the Chinese to figure out that they had invaded and seized control of Ryukyu, fearing that it would end trade with China so they allowed Ryukyu to maintain a great deal of independence.

Portugal and Spain were more focused on the spice islands, particularly those what is now East Indonesia, called Maluku Islands (at the time called Moluccu). These islands were, at the time, the only source of cloves for the entirety of Europe, and much of the world.

Holland's trade with Japan continued unabated.

I'm not feeling well at the moment so I'll stop here.

I'll follow up any questions you have later.

Cheers!