You have a shot but I suspect what will happen is you'll lose it in the last 4 miles. I went into 2 Chicagos in a row thinking I was in 2:45-2:46 shape and ended up with 2:41 at each of them. But those years I had 6-8 weeks of 85-100 mpw leading up to it. For sake of comparison, I died at Boston this past spring and did a 2:49.
Take the first 2 miles easy. When I did 2:41 I eased out in 6:22, and 6:18 before suddenly dropping down to 6-ish pace. I feel that those few extra seconds bought me fewer minutes later on in the race.
There is no shortage of people who can do longer runs at faster than marathon goal pace (e.g. your 7 miles at 5:38 pace) but the marathon isn't about how fast you can go for shorter distances, it is how long you can make the energy you need last at marathon pace. Without serious endurance (that comes through mileage) you pay the piper seriously in the last few miles. I hope you avoid that but I am not so sure you will.
For future reference
1. You are not doing very much mileage. If you want a faster marathon, you should probably have more and the extra miles should be easy.
2. I see no mention of any running at marathon goal pace (in your case, about 6:06 per mile). It's all well and good to run faster than your goal pace but you have to practice it too.