Neither of these are my field, but the situation reminds me of wildlife conservation. I wrote about that, here, a while ago. To the best of my knowledge, entrepreneurship is the only sustainable funding model that has been discovered for just about any purpose we can name. Free market environmentalists have a saying, "If it pays, it stays." One example of a self-funding conservation site is Hawk Mountain, not too far from Philadelphia.

So the question becomes: "How can museums raise revenue from their art while allowing visitors to view it for free?"

Even when the government pays for something, the value was ultimately created by someone who produced a product or service. So, if I ran the museum, I would not like to have the government play middle-man. Instead, I'd be thinking about premium products (raffles to meet the artists, art lessons, gift shops, etc...) and opt-in gameification. For example, something like a scavenger hunt in the museum. Visitors could pay a fee to join the game, and the first person to find the answer to a question about some obscure characteristic of a piece of art work in the museum wins a prize, with a new challenge each day (or hour, or whatever...)

And of course, there's Steemit. ; -)