Hog, it looks like your previously mentioned feelings of guilt just for using air-con and heating are probably misplaced since you don't seem to have a high carbon lifestyle at all. So I wouldn't let this stuff bother you too much or leave any sleep over it, I don't see the point. You seem to be very humble and thoughtful and caring.
That being said.......about funds...Many (most?) funds have at least some of the money invested in fossil fuel companies.
I chose 3 ethical/sustainable funds instead (via Fidelity). If anyone wants to look them up, I have:
FP WHEB Sustainability Fund Class C Acc
Impax Environmental Markets
Landlord Account M&G Positive Impact Fund I Acc
Since I invested in them, I think I slightly beat the stock market average.
Bank accounts are more difficult because almost all of them, especially mainstream ones, are invested in fossil fuels to a greater or lesser degree.
Particularly bad banks in the US for fossil fuel investment include JPMorgan Chase, Citi and Bank of America. For the UK, Triodos and maybe the Co-Op are free of fossil fuels.
I wrote to my pension provider Standard Life in April and they have investments in Shell and BP so I'll probably change pension fund at some point.
But the amounts matter. Someone who rarely goes above $1000 in their bank account and has no savings is contributing a miniscule amount to fossil fuel companies, while someone with $50,000 savings really could do a lot of good by being thoughtful about who it gets invested with. As usual, it's the rich rather than the poor that are causing the most climate change.