hmm...so it sounds like the key is having an actual proper Japanese military rather than outsourcing the soldiering business to feudal lords.
Pretty expensive that though, and hard to justify...have some minor conflicts with Korea? A stronger and more challenging Ainu?- perhaps we could introduce a WI I posted yonkies ago. WI the Ainu had 'civilized' and developed a country.
After the Japanese debacle at the Battle of Baekgang, Japan pretty much lost most of its desire to be involved in Korean wars. In fact, Japan built a series of forts because it was worried about an invasion from the Asian mainland, though fortunately, such an invasion did not materialize until six centuries later.
Sounds like the problem is that we need to give the warriors someone to fight. Centuries later, when Hideyoshi wanted to control the samurai in a newly-united Japan, he decided to just throw them at Korea and China (although they never did get to the latter). Heian Japan was a little less warlike than Sengoku Japan, so this doesn't need to be an attempt to outright conquer everyone. Just a little more involvement with other peoples.
Perhaps we can start with Japan building a strong alliance with one of the Korean kingdoms - I think they were already in good terms with Silla in Prince Shotoku's day - and then become militarily involved in Korea protecting their ally's interests. They might later become more strongly involved... perhaps some sort of diplomatic marriage and union?
Paekche might be a better candidate for an alliance than Silla. A few decades after Prince Shotoku's death, Japan fought on Paekche's side against Silla, seemingly motivated by worries about Silla dominating the Korean Peninsula. However, there were prominent Japanese families claiming descent from families in Silla, so if some of those families became more prominent, it could help improve ties between Silla and Japan.
On the other hand, if there's another traumatic equivalent of the Battle of Baekgang, whether it's in support of Silla or Paekche, Japan might decide its energies are better spent at home. Also, the politics in Korea at this time, as well as participation by China, would affect Japan even more, depending on how things play out on the battlefield.