Ideas for annual cover crops with allelopathic interactions that kill/inhibit common bermudagrass?

Good morning, I'm in the Coastal Plains of NC working with a recently reclaimed patch of woodland area.  

The spiny amaranth, pokeweed, and poison ivy have been difficult to deal with, but we're making headway slowly.
However, the common bermudagrass seems to come back thicker and thicker each season.

We intend to use this small plot to feed ourselves and our chickens.
So we've been using fire instead of chemical herbicides for weed control. But this method does nothing to harm the bermudagrass.

Based on just a little reading of negative allelopathic interactions, I'm considering overseeding with annual ryegrass or winter rye this fall. Then after manual removal of that rye, I'll follow up with seeding sorghum in the spring.

How likely would this combination help eliminate common bermudagrass on this plot after one year? Will I have to do this multiple years to remove the bermudagrass with this method?

Are there better alternatives to accomplish this goal?

I also compost in addition to feeding my birds. If I can accomplish multiple goals at once (like feeding the soil microbes or inhibiting root knot nematodes), that's even better. But my priority definitely remains common bermudagrass removal.

My zone is 8B, with high humidity for 8-9 months of the year.  
The plot is full sun (8+ hours), with soil pH between 5.0-6.5 depending on the specific sand/red clay ratio in a given spot.
Irrigation is not a problem at all, since this area holds water well (good organic matter), and it's downhill so it receives all the runoff (there's a swamp 100 feet behind it).

Thank you kindly.

The woodland that you "reclaimed" was doing the best suppression of bermudagrass you could have hoped for.  I dealt with this when I lived in Georgia for ten years.  Nothing I did or planted suppressed it for long.  Not repeated tillage.  Not penning chickens or pigs on it for months.  Not cooking it all summer under clear plastic.  And not growing other crops, although planting them after thorough tillage often enabled those crops to grow just for that season.  Since I was on principle against a lot of tillage, I ended up relying on cardboard and paper mulch, either around plants or planting plants through it.  This enabled me to grow a lot of stuff, but I had to apply the cardboard every year, except in small raised beds where the grass could be controlled by hand.  But I learned not to have any kind of permanent edging on the beds...the grass rhizomes get up next to or under this and then it's impossible to get rid of.  The whole area of mounded beds benefits from going under cardboard from time to time.
     That said, I eventually observed something....why doesn't it grow in the woods?  Or even up under thickets of privet and blackberries and whatever else grows when you leave a patch alone for a few years?  Bermudagrass (and it's partner in crime, nutsedge) is generally intolerant of shade.   Just as I was getting ready to leave that farm, I was developing an idea similar to swidden or shifting cultivation....going in a clearing a garden area out of "bush"....head high or taller weeds and brush....and then planting in that for a few years till the weeds found their way to it, and then moving to a new spot.  Fortunately in that situation there was more than enough land, and I had access to a heavy bush hog mower at need.   This tells me that maybe your cover crop idea could work, but it will take multiple years....let the cover crops reach full height, then mow and till and plant immediately with the next one.  You could mix clover with the rye and something like soybean or cowpea with the sorghum for some nitrogen fixation and even more shade.  I think the shading action is what you want....the challenge times will be when you switch from the winter to the summer cover and vice versa.  A thorough tillage might give your seeds the jump over the sprouts coming from the buried bermuda rhizomes.  
   Or you could do like I did before I realized it won't grow in shade....just plan to import and use a large amount of cardboard and paper every year.

So, depending on your timeframe, you could combine some of the more popular methods.

1) Solarization will kill the surface grass but not the deep roots
2) Sheet mulching is effective in the short term, but decomposes and gives way after a year or two
3) Occultation/Smothering works well. This can be done using black plastic instead of clear plastic, and is improved by using cardboard beneath the plastic. You can also simply lay something like a triple layer of cardboard and then cover that with 4-6" of much, basically a super thick sheet mulch that will last a lot longer
4) Tillage and raking helps remove a lot of the stolons and some rhizomes, and can be beneficial before solarization, sheet mulching, and/or occultation/smothering

So, if you tilled and smothered it during spring, then cooked it during summer, then tilled and smothered it during fall, then super-sheet mulched it over the winter, and then planted in the spring, you'd have a good advantage over the grass for some time.

If you wanted to permanently rid yourself of the grass (assuming no invasion from outside the area, you'd probably need to run this repeated occultation/solarization regime for three years.