A 5 minute timeout for Battle Arena when matches last 5 minutes seems pointless! Incidentally there is a timeout for AFKing in the lobby of Battle Arena blocks, just not games.
I understand from swiki there is a small penalty system in place for users that disconnect mid-game multiple times in a sort period of time. Courtesy of Google Translate:
If you discard or leave after matching, the matching rules will change, and only players who discard or leave will match. Eventually, it will remain 1/12 and will not change at all. It will return to its original state after a while.
AFKing or leaving during Battle Arena also counts as Abandonment, which is now a category available to report. The "Nearby Characters" menu is greyed out, but I believe you can exit to lobby after a game completes and use Personal Data > Communications Log > Players You Have Formed Parties With. If so I recommend you do this rather than selecting them with the mouse during a match as that would require you to temporarily abandon the match as well!