Another thread said that Apple identified this as "an emerging issue" in late December.
There seems to be no resolution yet, since the woman I'm helping with this issue purchased her phone yesterday and it is completely up to date. Her problem is almost identical, except that after clicking Sync Now, it would re-pause within a minute or so (not within seconds).
I know this seems silly, but maybe just keep trying it. She had already tried it several times herself before she brought it in for me to look at, and then I clicked it a few times as I was searching forums for a resolution. It has randomly decided to sync now (we're over halfway, so I don't foresee it stopping now).
I'm leaving it on that screen in the iCloud iMessages settings, and multitasking to Messages periodically view the progress, and then returning to the iC/iM settings screen. Truth be told, I don't really believe that whether or not you stay on that screen really has an impact, but I'm trying to limit the variables.