Nope, nope, nope. No fertilizer for carrots unless you want funky looking carrots. Seriously, they will have legs and arms and be weird. They will look fun but the usability is awkward. You would be better off getting a deep, decorative container and fill it with loose, sandy soil blend. NOT Miracle Grow garden soil.
If the soil is that horrible, you could also do raised beds.
And get seeds something like Tendersweet or Lady Fingers. Those stubby Nantes type carrots are better for livestock.