Coming from a Video Engineer the problem is not "take this digital signal, make it analog" but you also have to down scale it as well. The best scalers from companies like Extron, Kramer, ProBel and others are on the order of a single video line (30 fps*1080 lines = 32400 lines per second). They reduce the width and height of this line, convert it to an analog signal and then pump it back out the relevant connector.
This all requires DSP's and clever maths, which pushed up the price. Sadly it is not something I can ever see a low price FCPGA could achieve.
http://www.tomshardware.com/news/Blu-ray-HDMI-HDCP-Digilent-FPGA,14105.html for an idea of what sort of grunt you need to push things around in the HDMI world.
The second issue is attempting to remove the converter from aircraft, this is almost as difficult to achieve as converting it. The data rates that HDMI is capable of is circa 1.5gbps for 720 and 3gbps for full 1080p. With RF signals you have to increase frequency to increase bandwidth. This usually has the side effect of decreasing range due to attenuation. Think of the difference in range you get with Wifi vs FM (and if you really want to play with distance, LW).
Not saying it cant be done, and I am sure Esben has a better handle on the mechanics of it but I thought I would try and brain-dump the issue as I see it in case it sparks up some eureka moment from someone.