This is a tough topic. As a nonprofit fundraiser who has worked with large museums, I know that the staff and members would be very supportive or divestment, but there comes an issue with fundraising that the boards usually don't want to get entangled with. Most of the 'big donors' to nonprofits are the very same corporations/owners that are part of the problem. See, they're the ones with the money.
If public museums were publicly funded 100% (most get some public money), or fundraising staff concentrated on small gifts from many more people, then it would put less stress on the problem. It is tough though to divest your staff 401k while you major contributors and board members are on those same companies.
Once you start down that path, things can escalate out of control, then you have no funding whatsoever. Even slowly increasing focus on (grassroots-style) donations can set off a PR nightmare, not to mention most fundraising departments are not accustomed to it.
I don't have an answer, but I am searching for one. I do know that ultimatums are not going to work here the same way other divestments are working. To the movement: is there a way to list who we're divesting of and who we get funding from?