The LogStash HTTP output plugin has a headers field that one can use to add custom headers to the http request.
Why isn't there a headers field for the LogStash HTTP input plugin?
Fine. I'll just use exec and get curl to do it for me. 
Turns out that I needed to move the http_poller plugin which supports headers, but there are other issues that are blocking me from using LogStash. See Bug in http_poller: Environment variables don't always evaluate with http_poller.
If you are expecting a fast response because you need it now, you may be disappointed.
The majority of people that use this forum, including the devs, are volunteers, so they have real life to deal with too.
That was overly snippy of me, and I apologize. Deadlines make me grumpy. In better news, I'm trying to convince my company to pay for a support subscription. 