"Dead or carrot" (でっどおあニンジン) is in fact a play on the expression that is well known to Japanese, "dead or alive" (デッド・オア・アライブ). However, the meaning of this English expression is often misunderstood in Japan, such that it's sometimes used similarly to いちかばちか to express a desperate "sink or swim" situation. So once you add "carrot" into the mix, the meaning should come closer to, to use another famous quote, "give me carrot or give me death". But since that is merely speculation, the safer choice is to translate literally.