I think you should use songs that incorporate hand movements, or movements the children can do. They are not looking for someone with a great singing voice, they are looking for someone who is upbeat and loud when singing with the kids. Using your hands, and body will show that you know how to run a circle time and get the kids engaged. You are not going to sing to the kids, you are going to sing and get them to interact.
Top songs I would suggest
1)Itsy Bitsy Spider
-This has a lot of hand movements, and it can be expanded to the "great big spider"(use your whole hand when saying great big spider, not just fingers), and the teeny weenie spider(use the tips of your fingers(like you're crumbling something), and make small movements)
2)5 little monkeys swinging in the trees(sang to the tune of 5 monkeys on the bed)
-Kids love this song, especially the "SNAP" at the end.
The hand movements are:
"Five little monkey’s swinging in the tree "
-swing your hand back and forth, hand facing down.
"teasing Mr. Alligator can’t catch me….can’t catch me "
-Put your hands to your head(like moose ears), and move them back and forth. Also make your voice sound like teasing.
"along came Mr. Alligator quiet as can be"
-Sing this part quietly, in almost a whisper. Put you hands together(like praying), and make it like swimming.
"and snapped that monkey out that tree"
-sing this loud and excited. Clap your hands at the word snapped.
-This is another easy song to sing and use your hands.
"Twinkle, twinkle, little star"
-Flick your fingers from your thumb, or just open and close your fingers.
"How I wonder what you are"
-At the word wonder, and shrug your shoulders.
"Up above the world so high"
-Make a circle with your hands and lift your hands above your head
"Like a diamond in the sky"
-Turn the circle you made with your hands into a diamond.
This person on youtube has a lot of great ideas and uses the hand movements to the songs.
I would suggest looking at how she sings and how she "performs" the songs.