I have a personal tool to download all my replays in the "last battles" tab and i use it to compile some statistics on my game (i mostly use it to track my timeouts, but i also used it to evaluate the accuracy of the ennemy detection in the fog on Wondev Woman).
You have to use 2 API to do that:
POST https://www.codingame.com/services/gamesPlayersRankingRemoteService/findLastBattlesAndProgressByAgentId. Body:
POST https://www.codingame.com/services/gameResultRemoteService/findByGameId. Body:
The first one is to get the id of all your games in the last battles tab. You have to know your agentId for that. Use http://cgstats.magusgeek.com/ and you'll have it
The second one will give you the JSON you are looking for.