I just got the card and I have a serious deliquency, student loan, from 6 years ago. No other negs, but do have many new cards in the past two years, and lots of inquiries. My score is under 700! I did not instant approval though, I got the 14 day message, which was a joke, since minutes later they sent me an email saying that I got the card. I was actually on the phone trying to recon it and didn't even know I got it. I did get a low limit of $1000, but hey whatever, with good use and moving over some of the limit on my Delta Sky Pesos card, I will be fine. With you low limits I don't think it will matter too much about the utilization, but certainly if you drop the utilization on all cards as low as you can it will boos your score. Amex can be pretty good with CLIs anyway if you use the card and don't abuse your other cards.
I got the BCP, since I figured out the cost of the card and spending meant that there is at least about $100 in benefit after the fee. I am not so interested in the Sallae Mae card from Barclays, since I already have a Barclays card and have not had much love from them since I got the card. Since the SM card is a World card I expect Barclays to be a bit more harder in approving me for than than AE would be, especially due to the number of new accounts. I can certainly wait for a while to even think about that card. I might have to close my current card with them to get them to be nice to me.
I think you should simply go for it and try. I think the few point you might add in a month won't really matter too much. And even if you are declined, then you can always be bold and recon it. (You might even be able to recon the woman too.) Amex has great customer service and some of their products are nice too.
It is by will alone I set my mind in motion.....
Always follow these rules: Only take a HP for a new account. Always use the best rewards card for that reward category. Don't close a card unless you know you really should. Never use more than 35% of a credit limit. Recon as much and as best you can. Use the introductory period to the best advantage. Get the signup bonus. Whenever possible PIF or balance transfer so you pay less in interest. Never give an excellent rating when it is actually the norm. Always look for a discount as more is always better.
Always accept candy from strangers because they have the best candy or from people you know have good candy.
Or approved for that matter... 
Starting Score: 648