if you want it stretchy dont worry about the felting.
measure the base of the perch, the width, length and depth
say its 10" x 20" x 2"
find the yarn you want to use, maybe some others have suggestions, i dont know what your preferences and needs are, but i wouldnt worry too much about what it is as long as its washable and not prone to fraying. encore chunky might be a good choice.
swatch a gage. that means casting on 30 or so stitches (if its the chunky) and knitting until it measure 4" in length and then count how many stitches pr inch you have. say its 3 stitches pr inch
3*10 = 30
3*20 = 60
so cast on 60 stitches (or 30 if you prefer!)and knit 10 + 2 =12 inches in length (or 20 + 2 = 22 inches if you did a 30 cast on) in garter stitch
then make another one
then sew them together, leaving one end open to do some sort of closure like a zipper (buttons would be cat toys so zippers are probably better)
i think a simple slip stich would be more durable, but i dont know if you want to do it. let me know if you would like to try adn ill write that out for you