Probably the hardest achievement tied to difficulty besides beating the final boss. You have to beat the entire fairground level without healing (I didn't pick up weapons either but it doesn't matter because your best attack is not tied to them). For the whole level use only the running attack
with Bud. Never face enemies head on, especially not many. Let the enemies surround your partner and run into them. You will fight a clown early on. He deals a huge amount of damage taking off 1/3 of your health with a running hit. If he even hits you once restart. The next hard part is the one with the small cars. Those deals huge damage as well. Never face them head on. Either wait Terence to take the out or running attack them from behind. You have to watch out for your life for two reasons: one you want to have some room for mistake at the end. Two the lower your health the more likely the AI Terence won't heal himself to save the food for you. So both of you will be on the verge of death.
At the end boss don't wait for his monologue slam the toy rocket in his face and running attack him twice (or more if you are quick enough). You have to avoid the bottom part of the screen. Terence will put the ship back. Repeat. I think the boss needs 5 hits overall.