"the MORE money new capital is contributing, the larger the slice of the pie they could ask for relative to how much money they're putting in"
It is true that the subscribers to the new shares want to get as big a slice of the pie as possible in order to exert as much control as possible over the companies.
However, the bigger the pie that the new subscribers get, the less the government "earns" from recap/release - regardless of whether it opts for warrant exercise, SPS-to-common-conversion, or both.
For example, if the new subscribers get 70% of the companies, the government, the JPS and the legacy commons would have to fight over the remaining 30%. The problem is that the initial market capitalization remains more or less constant at about $250 billion.
So if the subscribers to the new shares receive too large a share, the government's share from recap/release will be reduced accordingly. The government gets diluted.
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