Recently I discovered the little X11 utility xclipboard (included with Apple's X11). This has made a big difference (and seems to have exposed a bug). It presents the current contents of the X clipboard, and will also keep a list of clippings that have been made. If I copy in Aqua, then click in xclipboard, then click in my target X11 app, I can paste there. This is pretty low impact for getting things done.
I have also noticed that very often xclipboard displays my Aqua clipping automatically, but then if I click one of my target X11 apps, the previous X11 clip contents get pasted in, superseding what I just copied from Aqua. But then I can get it back by hitting the [Prev] button in xclipboard, and use the same data anyway.
Maybe I have my X11 set up wrong, but even if I do, I can fix it with xclipboard.

