I figured out how to beat Pogo Party w/o using Catapult Plants, Mushrooms, or Coffee Beans!
Here are a few images. Hopefully the strategy and seed lineup should become apparent. Oh, and the instants I didn't have to use. Just don't forget the Garden Rake.
A good start.
Start of strategy completed. Now, to move on to adding the primary attackers...
The completion of the full strategy has started.
Here's a pic just to show that this build can complete the minigame.
Aaaaaaaaand we're done! Just to be clear, my pickup is Split Pea, Tall-nut, Imitater Tall-nut, Starfruit, Sunflower, Twin Sunflower, Flower Pot, Cherry Bomb, Squash, and Jalapeno, though the last three aren't really necessary.