peep
A palindrome of a very young child could be "civic," which reads the same backward as forward.
A simple palindrome for a very young child could be "mom" or "dad". It will be easy for them to grasp and remember.
Dove as a bird has the very short sound for 'o', dove as the past participle of 'to dive' has the long 'o' sound.
Doves and sometimes pigeons make a sound close to that description. Owls make a sound similar, but their call is more of a 'hoo' sound or 'hoot.' If the bird makes a 'cuckoo' sound then it is a cuckoo bird.
Wet stew is a palindrome for soup left out in the rain
racecar
The picolo is a very small wind instrument that has a very piercing sound. It is used in marching band as well as some performance symphonies in order to achieve a bird-like sound.
senile felines
Some options:laminate pet animal (doesn't sound very kind!)top spot (could be flat on top!)level (an all time favorite!)
race car
A little "chick chick" noise
You go Aha! but make it sound very very very very very squeaky, unrealistic, scratchy, lame, stupid, bird like fart noise.