Major scientific institutions don't get funding for saying "Things seem to be going ok right now", they get funding for saying "Things are screwed up, here's why, and we think we can fix it". I say this as someone who works at a major scientific institution.
We have an Institute for Creation Research [1]. There is plenty of funding for anti-climate change research. It's just not producing anything. That should give a hint as to where reality lies amidst interests.
I doubt that funding is even a tiny fraction of the research money from governments. And in addition it is money that might taint your reputation for future research funding rounds.
This is true in many areas of research. If you want funding there are certain opinions and buzzwords you should include when applying for funding.
Why yes, they do. Look at the LHC or major astronomic observatories - it's not like they fix a big existential problem for most people for the time being. Where do you work?
Could you name one major scientific institution related to climate science that dispute anthropogenic climate change?