This application will be delivered to external clients. Do I have to consider proxy implications from their side when they are running my application?
I would think that the client machine's network behind a firewall will it allow outgoing traffic to make outside contact with an external resource would be the consideration.
If so I don't find that much info out there about designing your code around this issue. I am worried that I am over complicating the issue by using things like HttpClient and HttpClientHandler etc. and then having to wrap all this stuff with a UI and further complicate my application.
Well in order to make contact with a Restful service, you are going to have to use HttpClient there is no coming around it.
You should try to implement some kind of SoC, like a Service Layer that holds code for the HTTPClint requests called upon by the UI.