If it works in Chrome, it probably means Construct is working correctly - if the problem was with Construct then it wouldn't work in Chrome either.
I think Brave has very strict privacy settings by default. This may mean it intentionally deletes local storage (confusingly also referred to as "cookies", although the term means any locally saved data). If it does that, then all you can do is change the browser settings to not do that.
Not sure if this will work but maybe giving the local storage plugin a "request persistent storage" option could be a workaround for stricter browsers?
The editor itself already has this option and it works fine for me on Brave. However, I don't know the requirements and limitations of this feature but I assume ideally we would have an action to request it and conditions to validate it.