You have already tagged the question with your answer. The best bid you have in ASP.NET Core is to go with Entity Framework Core. Entity Framework Core can take care of your most intensive data access requirements, and LINQ can take care of backend query generation (can use stored procedures too).
Check the tutorials here, EF Core is merely a bunch of C# APIs, nothing advanced needed to explore the runtime apart from LINQ. Overview - EF Core | Microsoft Docs
]. Your engineers will not have a tough time learning these.
If your users are already experienced in Enterprise Library then they already know how .NET framework can be used in enterprise environments, .NET Core is just a way to use-libraries-that-are-needed. Also, I just found (with a quick Google) that you still have some support of Enterprise Library in .NET Core, asp.net web api - Is there a way to use enterprise library in .NET Core 1.1.2 web api - Stack Overflow