One argument would be that the more money you have control of the more you can direct it towards less carbon-intensive aims. (Of course this is also an argument the effective altruists make, but they all turned out to be much less than altruistic.)
You assume that if you put your higher salary into savings banks will use it for evil, but not all banks/building societies are equal. At one end of the spectrum is HSBC/Barclays, but at the other end of the spectrum there are cooperatively owned building societies with the explicit aim of funding projects for the social good.
The other alternative is to use your higher salary in a different way. Maybe you can trade it off for time. I.e. if you take a salary increase but decrease your working hours so your pro-rata income remains the same, you have then freed up time you can spend on volunteering/sustainability projects/growing your own food/etc.
One of the drivers behind the correlation between wealth and carbon impact is that they will be using their investments to seek the highest returns without regard for social consequences, which as you rightly point out, will be carbon/energy-intensive.
But the other driver is more direct - high levels of consumption. Richer people use more energy at home (swimming pools, higher heating levels, more demand for cooling), more for transport (cars, multiple cars, flights), conspicuous consumption (lobsters flown in from Maine) and so on.
You're in control of what you spend your money on so if you feel you can live the same lifestyle as you do now despite getting richer at least the consumption part of your emissions won't increase.