What is the best way to use F# in a web development situation in order to make the most of the language? E.g. on its own with something like the Websharper framework, or together with C# and design tools, or otherwise.
It does depend on what kind of web development you're doing.
If you want to "make the most of the language" then I think WebSharper will do this - it will ease a lot of issues and make developing web applications more similar to developing non-web applications.
But, if the web development is not that complex then sticking just with F# and C# may be better than adding a major dependency on a package like WebSharper.
Currently, the C#/F# combination route might be more effective simply because of the maturity of the tools and example code available for C#. That is, use C# where the tools exist, but use F# for underlying logic. Since both are .NET languages, it shouldn't be too difficult to factor the modularity. Then, as better tools become available for F#, more of the project can be migrated in that direction.