I don't suppose it matters much at this point, but, have you tried a UI-based IRC client (like Colloquy for Mac or X-Chat for Linux)?
I don't know about any clients on Linux, but, Colloquy (and most IRC clients for Mac, I suppose) use a WebView from WebKit to display text, so they'll have points 1, 3 and 4 covered. And point 2, if you don't need chat history.
(disclaimer: I contribute to Colloquy for both iOS and Mac.)
Colloquy is the client I use when I'm on IRC, so I appreciate the work you've done on it.
Point 2 is actually the most important point for us. Since Tom was the only one that liked irssi, the rest of us lost the backlog and that was a deal-breaker.
Web-based chat, by its nature, is also very distracting. One click and you are reading HN instead of your internal chat room :)
Anyway, when I worked at an all-remote company, we just ran our own IRC server. Then all the channels were work-related. Of course, it was pretty easy to also join other networks, which we did (but would do anyway).
Oy, I dunno. I actually use an IRC gateway for XMPP group chats because irssi does so well for the medium. My team at work uses an IRC channel for adhoc group conversations, and that's the only IRC channel most of us are in. I also have a separate irssi open to freenode, but that's unusual for my team, and the freenode irssi is tucked in a corner where I see it only during interrupts anyway.