I bet the head of Sumeragi cannot refuse the offer because they realize of competent Harold is when managing territory.
In their mind, they were looking at the future when he is the head of Stoke Family and what kind of influence he will be making at.
Its hard to imagine the future, despite he offer the immediate and long term solution to Sumeragi family, the head cannot fathom what is the stake of Harold in his endeavor in this agreement.
We know he only wants to be alive by removing ahead the death flag, but for the Sumeragi Head, there's much more was going on and involve politically so instead of refusing because it is a one sided deal with him as the winner long term, its make him a real deal anxiety because he cannot read the intention of Harold, even his spies are not gaining any info out of it...