That is an excellent idea. Montreal already has something similar—the Montréal Museums Pass. I worked on that initiative when I was on the executive committee of the Board of Montréal Museums Directors, which brings together the chief administrators of 36 museums in Montreal. The board decided to try different marketing strategies in order to bring people into the museums. One of those initiatives was the Montréal Museums Pass.
This strategy is somewhat similar to the one you described, that is, for each institution to be able to sell the Montréal Museums Pass and obtain a percentage of the proceeds on the cards they sell, so as not to lose entrance revenues.
It benefits, too, since we know that everyone who buys the pass will visit these institutions three times. In other words, each card will generate about three visits, which is good. Furthermore, this encourages people to visit other institutions nearby that they would not necessarily have visited otherwise. However, since their entrance is free, it is good for them, too.
This could eventually be a model that others may want to emulate, since this pass also has a partnership with the Société de transport de Montréal. People can also buy a pass that includes public transit for a period of either three days or two weeks or even a year, if they want.