I would suggest removing the grass no matter which species it is.* Since you have such a small area to deal with, I would suggest you dig out the grass with a flat shovel.* Then you can sink the first layer of boards into the ground so the grass doesn't invade the area.* I'm not one to use chemicals such as RoundUp.* I've also read that RoundUp doesn't work on Bermuda grass.* From this site with more info.
You can try and id which grass you have at this site.
Using RoundUp to kill Bermuda is a bad idea but so is leaving the Bermuda grass. It is the worst weed you can have in a bed. Plastic, both solid and the so-called weed-blocking kind, is also a bad idea because it doesn't work well and fouls up the most important part of the soil - the surface just under the mulch. That's where the temperature is ideal, minerals are available, life is transitioning back and forth between the mulch and the soil, etc. The best method is to remove the top 1 and 1/2 to 2 inches of soil and toss it in the compost pile. Then apply the compost and other organic amendments. Do not till the area first - that drives pieces of the stems (rhizomes and stolons) down into the ground allowing it to come back forever as a horrible weed.
You might also get some help from this site.* Click on each grass on the left.
Do be sure to get SCREENED topsoil.* A mix of 60% SCREENED topsoil to 40% aged compost would be great for your veggies.* I don't recommend covering the soil beneath what you add as that will be a barrier to root expansion of your veggies.* If you like you can put down 6 layers of dampened newspaper as a weed barrier to stop any seeds from sprouting.* The newspaper will decompose over time.* If you dampen the newspaper the first breeze that comes along won't blow it around your yard.* :)* Been there, done that!* :shock:
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.