First, I would strongly recommend @Roberalz 's suggestion of using our Chinese support section. While that does exist here, that doesn’t mean that those reading the English support section read Chinese very well at all. I certainly can’t. So even where you link to Sogou’s website, it’s a challenge to me to figure out where to download the Debian package.
The sad, sad fact is that nearly all of the Lubuntu Team’s developers are native English speakers and have never had the need to switch input methods or keyboard layouts. That’s not a technical problem, but a cultural one, and it’s a doozy! We’ve been working really hard to add additional support and I’ve been really delighted by that but if those of us with the advanced knowledge necessary to answer difficult questions don’t have a compatible communication methodology, that doesn’t work so well.
I will also add that the Lubuntu Team guarantees support for what we ship, i.e. what comes with Lubuntu and what’s available in the Ubuntu archives. Anything outside of that is ultimately beyond our scope. We can’t guarantee compatibility with anything we’re not already packaging. That said, I think your really best option is to either inquire with whomever is developing Sogou or perhaps with the Kylin developers. Kylin uses a different desktop environment but I believe they use fcitx or at least should be familiar enough with it to help. Note that any advice you give for other input method managers (like iBus) are likely not directly applicable to fcitx and may be frought with other issues (a long time ago we used iBus and I remember problems).
Additionally, I will mention that if you’re using keyboard shortcuts, which I think is a common thing for people to do when switching between input methods or keyboard layouts, one thing you have to be cautious of is all of the other shortcuts. Are there conflicting shortcuts defined in the global shortcuts or in Openbox or in the application you’re working with? These are all important questions to answer before making any conclusions about how well keyboard shortcuts are working.
Recently it was brought to my attention that there seems to be some sort of bug in the keyboard layout shortcut handling for some layouts. There could be a similar bug with fcitx, but as it has a broader scope than LXQt (only a few distros use LXQt but LOTS, all with different desktop environments, use fcitx), I think the likelihood is much less. One thing you might want to do is only use the system tray icon to switch because then you eliminate any likelihood of shortcut conflicts.
Of course, that may be what you’ve already done. Meanwhile, I’ll try some experimentation and see if I can figure anything out. I hope that the information above will prove helpful and eliminate the need for it!