The WG cares far more about its own reputation that it does about the marines', the Gorousei made it clear to Sengoku and Sakazuki when dealing with the Impel Down and Doflamingo issues.
It wasn't a matter of reputation back then. Doffy threatened to leak information about the MJ treasure. They let the marines take the blame because the stakes were too high. The comparison doesn't work here.
And as far as we know they have far greater concerns that dealing with a reputated but otherwise still secondary marine captain, that isn't irreplacable to them.
Two Yonko are fighting a war, and all they did is send fleets to capture the Shichibukai despite knowing they can't capture them. They kept sending ships after Mihawk despite knowing how futile it is just to show the world they were trying.
The only way they could ever agree to make a deal about Coby is if they feel he's really an extremely important asset, or do think he has vital informations about them that they can't reveal.
No, that's not how it works. The WG cares a lot about reputation. That's the whole idea behind these cover-ups. If they don't care, then they might as well stop covering things up. Coby is a hero for the common folk. He's quite famous already. They can't just put this aside because other big things are happening.
And Sakazuki is extremely hardly someone who'll make deals with pirates, we already saw that with the Bonney-Blackbeard situation after Marineford, with Bonney being strongly implied to be far more important to the WG that Coby currently is.
Akainu was firmly against losing the Shichibukai. He's the last guy you wanna use in this kind of argument. He also cares a lot about his reputation as the head of the marines. So if he doesn't do anything to save Coby, a hero who saved many lives, then he'll be seen as incompetent. You can't protect the world if you can't protect one of your own.
The WG doesn't ignore smaller issues because they have bigger issues happening. They sent marines to capture the Shichibukai when two Yonko are fighting a war.
Coby isn't big enough to be their top priority. That doesn't mean he's not relevant enough to be a priority. This is bad reasoning.