The regular API is simpler than the RPC API, but it only allows you to download blockchain data in a different format. Verifying a message is a bit more complicated and requires input (the message and signature), so it isn't available in that API. I suspect that there's an issue with the way you integrated the C# wrapper (which is for RPC), and I don't think the RPC API is accessible via web at all, though I may be mistaken on that count.
From looking at the documentation for Bitnet (which seems a bit old, using RPC version 1.0, though I may not have found the right one), I'm not sure it supports SSL, or the verifymessage command. Have you tried using "http://" and port 80 to see if it works without SSL?
Well I'm trying to do this all online, not running bitcoind on my computer, and I think that's where the problem is. I've also tried it without bitnet.
If I have to do it running bitcoind on my computer, then I'll have to approach things differently (which is fine).