You can use the type system to your advantage. Cut a new type and inject a ReadOnlyDataSource or a SecondDatabaseDataSource or whatnot. Figure out what should only have one instance in your app, wrap a type around it, put it in the singleton scope, and inject that.
This has the advantage that you don't need an extra framework/dependency to handle DI, and it means that dependencies are usually much easier to trace (because you've literally got all the code in your project, no metaprogramming or reflection required). There are limits to this style of DI, but in practice I've not reached those limits yet, and I suspect if you do reach those limits, your DI is just too complicated in the first place.
* Your framework gets so complicated that someone rips it out and replaces it with one of the standard frameworks.
* Your framework gets so complicated that it turns into a popular project in its own right.
* Your app dies and your custom framework dies with it.
The third one is the most common.
At some point this stops working, I agree — this isn't necessarily an infinitely scalable solution. At that point, switching to (1) is usually a fairly simple endeavour because you're already using DI, you just need to wire it together differently. But I've been surprised at the number of cases where just going without a framework altogether has been completely sufficient, and has been successful for far longer than I'd have originally expected.
The answer is scope. Singletons exist explicitly for the purpose of violating scoping for the convenience of not having to thread that dependency through constructors.
https://testing.googleblog.com/2008/08/root-cause-of-singlet...