Does your bouncer let you view the full scrollback from multiple clients, without having to keep them constantly connected (and preferably without lagging out one's mobile client when it connects and the bouncer deluges it with old messages)?
There are a few different solutions for this out there, especially clients that run some custom protocol between the GUI and the bouncer, like Quassel and Smuxi - but I've tried both, and each both (a) has poor mobile support and (b) once you get used to it, turns out to just suck in general. (These opinions are a few years out of date, though.)
Thankfully I don't actually care much about mobile support: I don't use IRC for time-sensitive discussions, and for recreational purposes, trying to participate in a real-time conversation where everyone can type several times faster than you is, IMO, a pain; I'm faster at typing on iOS than I used to be, but it's still just not comparable to a real keyboard. Better to save the chatting for when I have one. But I find ssh+screen an unacceptable solution: partly because of issues with notifications (i.e. out of the box, there are none) and copy+paste, though both can be fixed in theory, but mainly because of the latency. When the letters I type don't appear for 100ms or more, it really trips me up and I make a lot more typos. I could find a server with less latency at home, but that wouldn't help when traveling, especially on an unreliable connection - even though there is no fundamental reason chat should be latency sensitive in the slightest. Tried mosh (ssh + prediction) as a compromise for a while, and it works better, but it has some issues and doesn't fully hide the latency.
These days I'm using Glowing Bear, the web-based remote for weechat (that uses yet another custom protocol). Latency's gone, and like other web-based clients it has the neat feature of embedding YouTube videos and images, so I'm finally pretty satisfied with my IRC setup. (And I can be paranoid about security and run it on localhost rather than using their website directly.) But there are disadvantages: Glowing Bear is somewhat feature poor, it (again like other web-based clients) is relatively slow to render, and there is no native iOS weechat remote. (Guess I could still use weechat as a regular bouncer, or perhaps try Glowing Bear from mobile Safari...)