I think it would be fair to describe the "REST API Client" generation tool in Visual Studio as "spartan".
This answer may be too late to help you, or there may be reasons why you can't use a different tool, but in the hope of benefiting you and/or future readers I'll detail how I achieved generating a REST client in NSwagStudio which reuses my existing classes and enums. (NSwagStudio is free and open source, and I have no affiliation).
On the left hand pane, we select our input. Besides the expected Swagger feeds there are some interesting options such as "Web API via reflection" which "uses .NET reflection to analyze ASP.NET Web API or ASP.NET Core controllers" - this is the option I used but my screenshot shows the default Swagger input.

On the right hand pane, click "CSharp Client" and switch to the "CSharp Client" tab.
The magic bullet is to untick "Generate DTO types":

This will cause it to generate only the client, so you can reuse your existing DTOs.
You'll want to specify a namespace for the client, and optionally one or more namespaces which will be added to the generated C# file as using
directives. For example, if you want your client to be in the namespace MyNamespace
and your model classes are in SomeOtherNamespace
you'd enter the following:

It's well worth having a play with the options. A few quick notes about some of the defaults and why I'm happy with them:
The HttpClient
is injected and you control the lifecycle (which seems to me a good thing)
A BaseUrl
property is defined. I haven't tested this yet but I'm hopeful from looking at the generated code that this will allow me to safely spin up multiple instances of the client class to talk to multiple servers which share the same API
The JsonSerializerSettings
property is protected
, but can be configured via the UpdateJsonSerializerSettings
partial method
I've saved my settings from the File
menu and added the .nswag
file to source control, so that I can easily regenerate the client in future if necessary.