Quite honestly, I do not know the answer to this question. The game itself does not suggest that you need to, but given the ample drops of carbon rods, it’s certainly possible to unlock Clownface (18,500) carbon rods.

Currently I’ve unlocked the Burns State Prison (7,000) and don’t foresee any problems reaching the next prize goal before the next issue unlocks. It might not seem like it, but 3½ days feels like plenty of time. Criminals in my town are plentiful and they each drop 15 carbon rods apiece. The numbers add up quickly, especially since the Superior Squad HQ generates carbon rods as well. This doesn’t even take into account the carbon rods you can earn by visiting friends.

If you visit your town often and stay diligent, you shouldn’t have any problems unlocking Clownface before the next part of the event starts. If it turns out you can still collect carbon rods once the second part beings, then you’ll be one step ahead of the game by having all the issue one prizes. There’s no downside to staying active during this event. Just don’t panic and spend donuts to rush things you’ll most likely be able to earn for free.

tl;dr – Maybe. Odds are that you’ll be able to collect 18,500 carbon rods (to unlock Clownface) before the next part unlocks in 3½ days, so make that your goal. If you already have the Burns State Prison, this won’t be a problem at all.