There might be a reason you've chosen not to do this, but you can make this game (and Perchance to Dream) playable within your itch.io page by changing 2 things:
- Set the "kind of project" parameter to HTML when editing the Itch.io project page
- Upload the zip again, but with any name other than the name of your downloadable version, and with the HTML file (little...0.0.hmtl) renamed to index.html
(You might also want to change the embed mode to "click to launch in fullscreen.")