You need a price. If you say, we need this infrastructure or technology and it costs x dollars, that can be justified, approved, and budgeted.
In most places I have worked, “my department uses something we get for free but they really want us to contribute what we can” would go exactly nowhere. Pushing too hard may actually even lead up a directive to switch to something less problematic, maybe even something commercial (that has a definitive price).
In my opinion, if they want this to work, they need to create a shared infrastructure for delivery that they can all use. This infrastructure needs to be a paid service for users with published pricing sorted into service tiers.
The base tier can be free with no support or “community” support. This tier can have a generous but finite usage ceiling. For higher volume users, there is a cost but also some level of “support”. That is, you can call somebody if the infrastructure is not working, performance sucks, there has been a security issue, accounts need to be segmented or merged, etc. You could also charge for performance. Why not both?
This service would operate as an independent company. It would be a service provider to the “foundations” or projects that use it. This means having payroll, legal, accounts receivable, support, and operations (eg. vetting the material they host). It would be a real company (non-profit ideally). However, instead of costing money, the service would distribute some of the fees it collects back to the projects it serves. At the very least, it would make the cost of distribution zero.
The most important part of the above is that there is definitive pricing for high-volume and/or high-need consumers. This can be budgeted and funded just like any other software or service purchase.
Problem solved.