For what you're trying to do, connecting from one web service to another service without using passthrough authentication it seems you'll need to authenticate usage of that service in another way.
So there's always indigo or WCF as it came to be called. Windows Communication Foundation is the enterprise 'serious' way of binding together enterprise services. it supports a range of protocols and authentication methods, one is authenticatino by certificate and basically that gives you the option of creating your own X509 certificate and registering on both endpoints (service access points) and then essentially you bypass the user credentials aspect by using that machine authentication method.
Web api is essentially designed to service clients requesting http or https. WCF can use a range of protocols for instance net.tcp which is way faster in terms of not having the http bloating ... don't get me started on thisone :))
Anyway, to continue in your case that your web api host authenticates to your other service with a certificate is hightly secure, found this article i think will help you out and if not the web is overflowing with examples.
Nine simple steps to enable X.509 certificates on WCF[
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There's also somebody doing this for web api requests and eventually succeeds, but i doubt that won't take a lot of extra work on your behalf to make happen, especially if you're using ajax
c# - How to use a client certificate to authenticate and authorize in a Web API - Stack Overflow[
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