Hi Conrad,
I’ve never encountered any issues but I’ve typically used good full-spectrum unicode fonts when I build kanji sites.
My best guess is that the font you’re using might be using an older encoding format which is tripping things up as you paste from the clipboard, or could even be missing characters which could confuse the renderer.
My Japanese isn’t solid enough to comfortably type much using IME, but copy paste from other apps where I’d prepare and translate was quite reliable.