We are trillions but you are just millions so give us your dilithium. Klingon children will go hungry if you do otherwise you selfish bastards!
Okay, let's break this down.
Is the dilithium on a Klingon territorial planet, the way the particles are orbiting a Federation territorial planet? In that case, yes "the Klingons" could simply come in and mine the dilithium.
Are the Klingon willing to relocate anyone who would be adversely effected by the mining process? The Federation/Sona partnership were.
Are Klingon children actually going to go hungry? If that is factually the case, and there is no other way of pervent this, then yes the dilithium should be mined.
Are the people in question indeed "selfish bastards?" If approached for negotiation, would they be amenable to trading for the dilithium, or would they be like the Baku and (probably) refuse no matter how many would be helped.
1,000,000 people are a lot different than only 600 people. A million people is the population of a small country, six hundred is the number of people in the stands at a high school football game.