Well, first of all, this has nothing to do with asp.net-core per se. This has more to do with how Dependency Injection works. You have to ask yourself why your method is static. Is that really necessary?
If you can't get rid of your static method, you might as well go all the way and introduce another anti-pattern, the Service Locator Pattern. In short: In the Startup class you put a reference to the ServiceProvider in a static property (call it for instance "ServiceProviderSingleton") of a static class (for instance "ServiceProviderProvider"). This way you can just call "ServiceProviderProvider.ServiceProviderSingleton.GetService()".
Again, i suggest giving your overal design a critical look. But if this is what you need/want then I hope it helped.
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