Hrmpf. Seems the Itch app is pretty much unreliably usable since like two months or so.
The issues you have with the downloads might be the same cause, so it will probably not work with app either. My hopes were that at least the app has some kind of download continuation, if things fail. It does handle downloads a little different, as far as I know. But I do not know how exactly.
This workaround worked for me: https://itch.freezing.top/post/14552731
I needed to click update Agent or something like that. And follow step by step. The Itch app is a chromium browser and whatever user agent it is sending is not working.