If you have a small animal vet in your area who is willing to help you they should be able to draw the blood and prepare it for shipping to Cornell. That would be preferable to hauling to the vet as that will cause the bloodwork results to be skewed.
As for Peanut - she's adorable but there is definitely too much of her to love so getting her diet dialed in is going to be very important in addition to making sure you have baseline bloodwork performed on her. A normal weight for an equine would have them in the 4.5 - 5 range on the Hennecke scale. A measurement of 6 is not normal and is an indication that there needs to be a weight reduction. Looking at Peanut's pictures I'm guessing she's about 75lbs overweight so if she's 950lbs now and her ideal weight is closer to 875 you want to feed her 1.5% of her current weight or 2% of her ideal weight, whichever is greater. That would be 17.5lbs a day TOTAL (Hay + concentrates).
I'm guessing that since you say she's in a herd that the hay amount is actually a guess, not an actual measured amount she's being fed per day, is that correct? As far as the grain, nothing Nutrena makes is suitable for an IR equine. As noted in your welcome letter we recommend under 4% fat and under 10% ESC+starch - the Special Care is 7% Fat and the last time somebody submitted it to EquiAnalytical for testing it came back as over the 13% ESC+Starch indicated. That result is here: https://ecir.groups.io/g/main/files/9c%20Analyses%20of%20Various%20Feeds/Nutrena%20Products/Nutrena%20SafeChoice%20Special%20Care%202020-09-16.pdf
Were she mine I'd treat her as IR even if you don't have bloodwork to confirm that. That would include no grass turnout, no grain except what is needed to get her minerals into her and a measured amount of hay per day. If she can't be on a dry lot she should be muzzled with a sealed muzzle when out so she can't graze.