Does something like:

CROW_ROUTE(app, "/add_json")
.methods(crow::HTTPMethod::Option)([](const crow::request& req) {
// here you can set the response code
});

not work?

will be closed as no answer to comment.

Sorry did not have time to investigate more.
On the website: https://crowcpp.org/master/guides/routes/ it says

Crow handles OPTIONS method automatically. The HEAD method is handled automatically unless defined in a route. Adding OPTIONS to a route's methods has no effect.

I did not try exactly the example you posted here, but I did try in a dynamic route and the handler was not called

It seems that the http Standard prefers 200 but also allows 204. (see https://www.rfc-editor.org/rfc/rfc9110.html#name-200-ok)

Currently 204 is hardcoded in routing.h.

As a workaround you could patch this.

As it seems that some browsers depend on it (see https://stackoverflow.com/questions/14675330/what-http-response-code-should-be-used-for-an-option-request) I would propose to add an Cmake Option "HTTP_OPTION_RESPONSE_RETURNS_OK" which defaults to off and to switch the return code based on that option.

Alternatively we could switch in general to 200 but may run into issues maybe with older browsers or clients which make special assumptions.

Can the routing. h file be modified to return the Access Control Allow Origin request header for OPTIONS pre check requests?

Yes, that modification could be done. But normally it should be supported by CORS Middleware.
see https://github.com/CrowCpp/Crow/blob/master/examples/middlewares/example_cors.cpp and https://crowcpp.org/master/guides/included-middleware/.
If that setup does not work, feel free to come back.