I don't see why you can't implement global exception handling and logging that can be invoked during WebAPI start up in Startup.cs
https://stackify.com/csharp-catch-all-exceptions/
Below is WebAPI implementing GEH and logging with Serilog and the ErrorHandlngFilter.cs stating at Program.cs
https://github.com/darnold924/PubCompanyCore3.x/tree/master/WebApi