I also use Avast and was having the same downloading issues. I think I may have resolved the downloading issues as per the following steps with my avast app:
Go to the menu; settings; general; blocked & allowed app > unblock the Brave app
Taking those steps seems to have fixed the issue for me.
I have used Brave for a long time and have always been able to download files on Brave that I want to keep to my desktop – until about a week ago, when a message came from Avast (free version) and I clicked “no” instead of “allow” – not understanding, it turns out, what I was doing. Since then, every single time I try to download anything, I always get “Failed - Insufficient permissions” message. I have tried to fix this by rebooting, re-installing Brave, using settings to change permissions, etc., etc., etc. Nothing has worked. I have to use Chrome or Firefox to get the downloads. Windows 10 is what is on my computer.