Secure a base of operations.
If Ceres and Vesta are not defended, the first step should be to take them. This will give you a base of operations and deny the enemy mineral resources at a time they would badly want those resources in order to build new hulls. The mining colonies not being defended would also be a thing that might be in a pre-war contingency plan so you can play up how the aliens have good spies.
Play up the politics.
Declare the mining colonies independent state under the protection of and allied to the aliens. The aliens are only here because the poor persecuted miners called for help to free them from the tyranny.
Hire some humans who look presentable on video to do your talking for you. The money is good and retirement package generous with relocation to another star system included so you'll have no issues finding people. For that matter, in many settings there really is genuine friction between the belters and the inner system so the belters might actually take you up on the independence.
Continue by promising to liberate Mars and Moon as well. While this might not make sense at face value, you are aliens so nobody expects you to actually know or care about human politics. And it does match well with your known actual goals of getting to the resources at the moon and cutting down human power so it is believable statement of intent.
The merchant aliens probably have the resources and connections to arrange some credence to these ambitions as well. Since they do not actually need to win election or arrange a genuine uprising just be visible in media, you'll probably be able to generate political movements in Mars and Luna for independence.
Again it does not matter if this makes actual sense. Majority of the things the brexiteers said during the referendum campaigning made no sense whatsoever and nobody really cared. People just don't. And this is not going to change any time soon. Politics is not about determining facts, it is about choosing sides. And "liberating the colonies" sounds much better casus belli than "securing our monopoly on a strategic resource for maximum profit". More importantly it also sounds much better than "breaking a monopoly on a strategic resource so that the elites back Earth can get more power and money".
Political support is a strategic resource just like metals from asteroid mines. Denying human governments the freedom to define the conflict as alien aggression that all humanity can unite against is an obvious thing to do. Since you are not interested in or capable of conquering anything there is really no reason to not dress up your war goals.
Nobody will believe it is the actual reason of course but it is believable that you will go away after achieving your goals and leave the colonies "liberated" and people won't really care about strategic resources enough for that to matter.
Send a fleet that is significantly larger than best intelligence thought you have towards Mars on a fuel effective (slow) trajectory. Make clear that liberating Mars is on. Challenge the Earth government to the decisive battle to protect their colonial empire, if they have the guts. "We are even taking it slow so you won't have excuses for not making it to the fight."
Play up how the belters who volunteered to liberate Mars make up a significant portion of "the Fleet of Martian Liberation". Encourage all sympathizers on Mars to step up their actions "so that Mars can be liberated without actual fighting and collateral damage once the space battle is over". Discuss with Martian government how the planet cannot be defended once you control space and the details of transferring political power to "Free Mars" without unrest and disruption.
Assure the Martians that the fight is just a formality. They will be liberated for certain in the imminent future because while the brave soldiers of the human fleet do not lack in courage and ability their ships are unfortunately obsolete crap that will only perform well as targets for shooting practice. The responsible thing for the human government would be to just surrender but unfortunately politics requires that the innocent soldiers must die before surrender is an option.
Earth really must take this challenge. Mars is not something they can hand over to you and if they refuse to accept your challenge here people will with good reason ask what is the point of having this war at all. And since you cannot possible have another fleet in hiding with the fleet en route to Mars already exceeding what you were expected to have it will even be safe to send almost all ships defending the Earth to the decisive battle. This is going to be their best chance to take you on so they'll take it.
Lunar smash and grab
The reason the Martian fleet is bigger than your force was expected to be is not because reinforcements ferried to your bases on the belt. It is because it is entirely fake. It is just a bunch of painted metal on a ballistic "fuel efficient" trajectory towards Mars.
Add some EW systems, radars, radio transmitters to create "encrypted" chatter, and small maneuvering drives to avoid collisions. Should be fairly cheap to build and the ballistic trajectory with the EW systems should make it look real enough. Not having full powered space drives is not obvious if you do not use them.
Once the Earth fleet is far enough with enough momentum you'll send your actual fleet for the actual objective at Luna. Since the humans should have seen no reason to keep lots of ships back you should have comfortable superiority. And they should have gone for fast transfer option for Mars so they should not have the reaction mass to get back to Earth in time. You time it right and they'll need to stop to refuel at Mars.
You will still want to time this so that Mars is far from Earth and your bases at the belt and Earth and your bases close.
Interesting aspect of tricking your enemy to move in response to your actions is that you are essentially controlling their trajectories since the timing makes the best trajectory for them to take predictable years in advance. The only variable is their decision making delay. So depending on the technology it might be practical to have a mine field waiting for that Earth fleet on its route to Mars. And on its way back too.
You have either succeeded in taking your objective on Luna or been defeated. Mercenaries do not fight to the death and you have committed no atrocities that would make surrendering a no go. The Earthlings have no real reason to treat you as anything other than the professional you are.
If you have gotten the objective continuing the war is pointless. Contact Earth authorities, tell you have the loot and your ships are in position to do orbital bombardment on Earth. Isn't it time to end this nonsense? I am not getting paid per hour you know?
Suggest immediate cessation of hostilities and ask for recognition of the independence of your friends in the belt and referendums on independence for Mars and Luna.
Realistically almost all of the human shipyards should be in the Earth system and you would have destroyed them already as legitimate military targets after giving the personnel a chance to evacuate but if this is not the case you'll also want to ask for dismantling of military capability. Your customers will probably have something more specific in mind. They'd probably want a copy of all human research data on FTL for example on the off chance fresh eyes noticed new things.