You can absolutely enter a WP class. But why do that to this horse? It's not built or trained for western pleasure and it's unlikely he would enjoy WP. That big trot stride which has you bouncing out of the saddle and the pinned ears when you tried to slow him down is not demonstrating a happy horse which is a "pleasure" to ride - the overall picture is what WP is about, not just how low & slow can you go.
Showing is playing a scripted part better than everyone else - we are paying for a judge to say "you fit the mold I like best." You will be out of place in a WP class; you will likely be flying past the bulk of other riders, potentially making your horse or theirs nervous or distracted. You will likely irritate your fellow competitors who have been working hard to succeed in this particular event, and you & your horse may earn a first-look reputation as "that crazy paint" rather than making new friends.
Show your lovely horse some respect for his unique traits and abilities by participating in classes that let him shine! Ranch pleasure, equitation, western riding, or trail may be better options for letting this horse try something other than speed events.
That said, I started my 4-H showing career on an inappropriate mount. We participated in exclusively novice classes so we didn't negatively affected more serious competitors. My little backyard grade horse tried her heart out for me, but she was so happy to return to her trails when I bought a show horse for the next season.