Synchrony uses TU FICO 8 unless it's changed recently.
Depending on how long you plan to carry the debt and how much it is. It would be better to grab a short term 0% credit card or one with a nice sign up bonus. Your scores aren't bad by any means.
BofA and Citi give 12 to 21 months on 0% offers. Problem is there's no guarantee on the starting limits being high enough to carry a $7k balance without hurting your utilization.
I'd suggest finding a less expensive ring. The money can be used elsewhere for things you guys will actually need. Part of being an adult is knowing the differences between wants and needs. And deciding the best course of action accordingly. Don't put yourself in a bind over apiece of sculpted metal and a rock.
The interest rate on the jeweler store card will eat you alive if not paid off during the 0% promotion. That's how they trap you and make money.
I suggest a short term loan as well - and if you can get one for 12-24 months same as cash go for it. The reason, in my opinion, that you want it to be short term is because, if you take out a long term loan and things go South shortly after you get married, you won't want a reminder every month for 72 months!!!!
Just kidding! 
Good luck to you.
I would suggest buying something you can afford now and then upgrade later. Like what everyone already post, it's not a smart move to get yourself put into a bind. If she loves you enough she would understand and appreciate what you are trying to do. There's always room to upgrade.
The next thing is to start save some money so you can put a down payment, or just save the entire $7G to go and pay it out right. If you are already set in your mind that you want to finance this ring on a credit card then go for the 0% term for 12 - 24 but with your school not sure about the limite you will get.
Sorry to say it but you have to decide what's going to be the best course for yourself.
Gardening 10-13-2017
I'm not going to tell you what to buy. If you want to splurge on something big to make a big impression when you pop the question, you're well within your right to do that, and it sounds like you've already made up your mind on it anyway. Buying an engagement ring is not something we necessarily need to be rational about.
However, what does matter is how you're going to pay for it. If you can handle paying down $7K in a year then you're in a good spot. Credit cards are an option, but they can be risky since you don't know if you'll get enough credit when you apply. Also, if something happens and you take too long to pay it, the interest will be punitive. However, you may have the option to get a balance transfer to another 0% card if you've paid down a good bit of it and your history stays good.
You may want to look into other short term financing options just to be sure you can cover the up front. If you have a 401(k), you can borrow from that at a pretty good rate, and there's no credit check. That's what I did for mine. You'd be essentially borrowing from and paying interest to yourself. Payments are deducted automatically, so you never see the money. It may negatively affect your fund's future in a bull market, though.
You could take advantage of the store's financing, which could be good if they offer a 0% grace period. However, you MUST be sure you can pay it off during that time, or have a backup plan to do a balance transfer after, otherwise you'll pay dearly. Store financing typically carries the worst interest rates on the planet.
Good luck!
