I have uploaded a ZIP that contains an EXE and other essential files, Is there a way to tell the itch.io app which file within the ZIP to run?

Edit:

It looks like when installing the game it unextracted the ZIP, but it still isn't running the .exe within the folder.

Edit 2:

Manually running the EXE, it seems that I can't run it more than once. (I use packr to build a .jar into a .exe)

I got It to work, I used Launch4j instead of packr.

Yes there is, you can add an app manifest. It doesn't always work perfectly, but should improve things.