You are close to qualifying for their charge cards but my guess is that you won't. You might get their counteroffer of getting a secured Zync card which I have heard some people getting. I don't think you need to wait a couple of years to apply. Why bother as a rejection will just generate a inquiry which will disappear in 2 years anyway. As FYI, AMEX mostly pulls Experian.
Some advice before you apply:
1) You probably have a few points more to get from getting your Discover balance below 10% of its limit so $350 but make sure one card reports something. Also, make sure you know when the credit cards report to bureaus if you are spending on them.
2) You need to talk to your installment loan and see if you can get them to remove the late payment. Many will do so and it will bring your score up a bit. Or maybe dispute it as well. Often they don't keep payment records that far back. Otherwise once that late payment hits 5+ years, you should see a little bump.
3) How long till the hard inquires fall off?
4) You aren't really getting real FICO scores. I find FAKO scores to be highly inaccurate in any case. If you think the info is the same on all the bureaus, then go to myfico and buy one real FICO score (I would say TU). You might be surprised. Also I find this tool useful. You can use it to try different scenarios:FICOÂ® Score Estimator
Once you get your FICO score and assuming it isn't high enough, go to creditkarma.com and have it monitor you. It will also give you a FAKO score, but don't worry what the score is (since you already have the TU real FICO score from myfico.) but pay attention to how it moves. You can update it daily.
In any case, if you can get FICO to above 680, you should have a really good shot. AMEX has its own scoring model based on information it pulls from Experian mostly. Don't bother waiting 2 years to apply. Just get your scores as high as possible in short term and apply (do it over the phone). If they reject you, call up the reconsideration number. Even in your current situation, you should be close.
Amex Centurion, Amex Platinum, Amex BCP 8k->24k (5/23/12), Amex TE 15k, Cap One 1.5% 15k->20k (8/7/13), CSP 25k, Chase Palladium 100k, Citibank AA 35k (AU), Firestone 1.8k->2.2k->2.4k (8/20/12), JFCU Jloc 30k, PenFed Plat Rewards 30k, SF Fire 30k, US Bank Cash+ 25k