I don't know if carbon will win or not. Right now, carbon is in a race with renewable energy sources (wind, solar, + storage), and their costs are plummeting while fossil gets more and more expensive. We're already at the tipping point where it's cheaper to build out wind/solar than to build fossil power plants, and it's much cheaper to fuel a car with electricity than gasoline over its lifetime.

The population thing is already on a trajectory toward leveling. The main cause of population growth now isn't birth rate, but rather increased lifespans of those born two or three generations ago. If we convert to a renewable energy base before population levels off, we win.

Various governments are still heavily subsidizing fossil fuels, and so it's an unfair game. How can renewables win in this scenario...

They won't keep doing it. The advantages of renewables are too strong.

The thing about politicians that can be bought, is that someone else can buy them, too. If they're not ideologically committed to the cause, if it's just political, then they'll turn on the cause in the name of either ideological goals or simple practicality.

There's a hard limit to how much harm a politician is willing to do to their own country in order to placate special interests with nothing to offer but money.

Lobbying and voting for pro-renewable politicians (or at least those that are neutral).