You can upload any type of file to Itch, including an installer with an MSI extension.
But as you've probably already read, this is NOT a good practice for two reasons. First, the Itch application that allows you to download and play games isn't compatible with installers, so your application won't be usable that way, only through direct download from the website.
Second, nowadays, if a game from an unknown source requests administrative privileges (even during installation), it's considered a red flag and is discouraged for security reasons.
You should seriously consider whether associating that save file with your application is worth the drawbacks. Personally, I don't recommend it.
PS: I'm not 100% sure, but I think you don't need administrator privileges to associate a file type within the scope of a user account.
PS2. Why don't you do what all the other games do and have a load menu within the application itself?