I feel like you would enjoy Slice and Dice. Similar to dicey dungeon, you are rolling dice, but instead of them being a classic d6, each side of the die is the hero’s abilities. You collect relics and get better heroes as you progress through the run. 8 bucks on Itch.io gets you both the mobile and desktop download. Also has a demo.
That’s how I got my kitty! He climbed out of the kennel onto my shoulder and sat there like a parrot.