Is there a specific paint brush that my Neopet needs to be painted with in order for me to pull the lever?

No, there isn't a specific paint brush that your pet should be painted with to pull the lever, but there are certain paint brush colours that won't work in the Transfabulator. The Transfabulator cannot be used the following painted Neopet body types: Maraquan, mutant, baby, invisible, and fruit and vegetable Chias.