I had it easier: I was trading with both of them and simply used a monk to incite revolts in their provinces - as soon the rebels take over, I move in and capture it. Rinse and repeat, and I was trading them till their last province fell to rebels
It took longer but it also gave me time to expand in the sea and capture all but one trade nodes, build a huge navy and become a huge trading empire; while all I had was one and half stack of land troops (many samurai though
)
So in my case, Shimazu were the master of clandestine operations