Wow, that's so frustrating! It's odd that they have criteria like that and that they were willing to share it with you. I've spoken with several underwriters at different banks and CUs before deciding whether or not to apply with them, and no one has ever shared their criteria so explicitly. It seems like they would look at more than just your score; though the 690 FICO may be their minimum requirements.
If any of your CCs are reporting balances and you pay them down/off, your score should go up after it posts to your CR. So, that's one option. If it were me, I would probably go ahead and call them on Monday and see if you may get further with someone else. It does seem silly that 1 point should keep you from being approved; it probably depends on how flexible they are and on how much leeway their underwriters have.
Good luck! I hope you get approved on recon. 