I've read where people have downloaded DCS Manager to their Droid phones, then sideloaded it to the Fire
, so I'm pretty sure the app is compatible.
The Fire is probably more Android compatible than you realize. It runs 2.3 (though an Amazon'd version), and every android app I've sideloaded has worked so far (incl. games like World of Goo, Anomaly HD, Edge). It can be connected to the Android Market if properly rooted, but being I've had the Fire for only a week, I don't intend to do this for quite some time (at least until my warranty is expired).
Besides, the 7" size is perfect for a kneeboard. I use it for my F4 PDF checklists, as you suggested.
EDIT: I just found it on GetJar.com, but it's not as simple as downloading the apk. They ask you your tablet model, and putting in the Fire tells you that it is compatible. Of course, GetJar also said that the EVO 4G is also incompatible, and we all know that this is inaccurate (else we wouldn't have a review