This is fun! The core concept is good and goofy. It's a really fun take on 'only one inventory slot': you must give up your power-ups to increase your score. That's good stuff! Having the player learn what the different hats do by picking them up is very fun. I found my curiosity sparked and genuinely wanted to see more hats and discover what they all could do.
This is probably unintentional but I found it really funny that the player carried the hat instead of wearing it!
Some more tweaks and a soundtrack and you'd have a really great lil arcade game.