/$tyk_context.headers_Host/
This is obviously not expected, which means that Tyk did not replaced context with value.
Pls ensure that in your API definition you have enabled context variables enable_context_vars, it is on advanced tab.
In general, trick with rewrites should work, because if you specify custom host name inside rewrite rule, it should rewrite host header too.
You can check it by setting “rewrite_to” to smth like https://test.example.com/$1.
Also pls ensure that your “host” contains HTTP scheme, so it will be smth like: “rewrite_to”: “https://$tyk_context.headers_Host/$1"