First, go to brave://settings/shields and set Shields to Aggressive, so it blocks first-party and third-party trackers/ads/anything like uBlock does.
Second, this is what I meant in my post, you had to start switching things off individually to know exactly what it is.
For example, set Fingerprinting to Allow Fingerprinting and test, it seems that fingerprinting protection could be the problem, even on Standard I got problems logging in and making some captchas.
You could also try going to brave://settings/cookies and add [*.]epicgames.com to Sites that can always use cookies with the including third-party cookies checkbox on. and try again.
But don’t think it is this reason, because Brave is not really blocking the cookies but putting them in the Ephemeral Storage, which means they are only seen by EpicGames, and used accordingly. But doesn’t hurt to try if issues persist.
It is not the Adblocker because it is only blocking tracking stuff, not really any essential scripts for the captchas.
When I got issues, Allowing fingerprinting, seems like made things better, sometimes not even having to input captchas and all that.