But does iTunes try to set the phone up as a new device or something?
I'd first check the folder where iTunes backups are stored (forgot) and ... backup that folder just in case.
Then if you are jailbroken you can via FTP copy the whole /var/mobile/Applications folder which contains each app setting and then set up the device as new then copy back the settings for each app, which of course can be a PITA unless you code some bash script to automatically do that for you.
If not jailbroken then you could do the same as before but instead of FTP then using an app to read the contents of the phone via USB (Don't recall names now) regardless of it being jailbroken.
The hard part here would be getting it back inside, but apparently some of those apps can write (doubt it, but never tried, since I'm always jailbroken) regardless of jailbroken or not.
I also, don't know if possible via the phone (i.e not depending on iTunes) but if you can turn iCloud backup and then, have enough space available for it (or $$$ for an extended plan), make a backup that way.
Seems to be the painless option if available.
Of course all of this just pretending like your issue is never resolved and you really, really wanna make a backup of your phone.
The message here for you to learn is, buy an external and have a backup of your OS X hardrive (or at least the most important locations) via TimeMachine (ugghghghghhhhh), CCC, SuperDuper, etc.
I forgot the amount of times I had to start from zero or do the first option in this "advice" post, just because for the reason it might be, I lost my entire Hard drive.
Now, you might ask why I said make a backup of your iTunes backup folder. I don't know if the info might be there and it's just a bug (you can check each folder inside that iTunes backup folder and then order the files by size and if you had videos there you might play with file extensions and see if the videos there correspond to videos on your iPhone, and so on -hopefully you get the point-).
This way you make sure it's just iTunes playing dumb and the phone backup is there.
What if it's there? Hard to say but probably I'd copy iTunes library, library catalog (the XML or ITL I don't remember which one is the important one, google) and the backup folder.
Then reinstall OS X and restore the iTunes library and hopefully you can get the sync working again.
Anyway it's pretty late and I'm sleepy so I just likely didn't make a lot of sense but take what you think you can work with, if anything at all.
Hopefully you can solve your problem soon.