Since Kate can open remote files, I think if I can configure a project (or, if possible, each file) and define a website to prepend the "source" string to, it will be great.

Why? Using lxr for example is too much for just getting the context of a string. I don't want to have my internet browser open as it just eats the ram for no reason, where I want to focus on fast translation work.
And with the new preview tool Kate is getting, we can also display Qt ui files from there!

Idealy, we can (for KDE projects) extract the results from the lxr search and show them in a dialog, to open in Kate. But I know this is too much, so I'm more than fine with the trial and error way :)

Of course this should not disable the use of lxr, as it's very important for in-file searching.