Finding a nature for a Pokemon - especially seeing that's a status/supporter type - would be hard to find a nature. But, I think is should be:
Impish vs. Careful
Looking at this, you need to eliminate what you're not using. "Sp. Attack and Attack" don't matter (as much) within your move set. The only thing it will go under is "After You" or "Copy Cat", based on your opponents decisions. Therefore, it's only up to luck.
The reason I choose to reduce "Sp. Attack" over "Attack" is because of Belly Drum - maximizes your Attack. So, attack should play the higher role in this situation.
Now, the boost section. Swirlix's stat's are fairly average - even Slurpuff's when evolved. So, it won't be much of a difference leveling up. Defensive strategies should play a higher role since you're not using Swirlix much for attacking. You would wan't defenses to withstand attacks when using any one of your moves - to prevent failure. Personally, I would choose to go more for the "Defense" side, rather than the "Sp. Defense" side. The reason is because of his weaknesses - poison and steel. 17/29 of the attacking moves of these types are physical (slightly more than half, but more efficient. Also, steel types are commonly used (lots of defense and resistances) and have 11/14 moves being physical.
His defense is still slightly higher, but this nature will definitely strengthen it. Poison types contain more "special" type moves. Therefore, it's up to your choice.
Normally, speed would play a big role because you might want to use moves like "Yawn" or "Copycat" much quicker, but the lack in speed can also counter to boost your chances in using moves like "After You". So, I didn't bother with speed.
Altogether, I would personally select Impish based on my reasoning, but your approach may be different. Your Choice! Hope I helped! :)