So- it looks like running client-side Blazor on a 'classic' IIS site as a static page works well! Blazor is starting to become something where someone besides a true, heads-down hard-nosed developer can experiment!
I managed to host the default Blazor template on my own site. The initial upload content looks a bit messy with lots of tiny dlls (which surprised me a bit), but the speed of the site is respectable.
My next task is to try and access my web API to see if I can call my image stores. I'm definitely refactoring my base website to use Blazor, since it's technically an SPA anyway.
You should give it a try here.