What do you do when one team forgets that it has an unplayed boule?
Team B had an unplayed boule. Team A gave Team B the opportunity to play next. Team B refused to do so. It makes no difference that Team B acted because of a mistake rather than a deliberate desire to cheat. For whatever reason, Team B refused to play when it was their turn to play. Team B was at fault.
The determination that Team B was at fault rests on the idea that (a) playing a boule out-of-turn, and (b) refusing to play when it is your turn to play, are both are violations of the rules. They are flip sides of the same coin. Just as a boule can be played contrary to the rules, a boule can be withheld from play contrary to the rules.
Note that when a team throws a boule (or withholds a boule) contrary to the rules, that boule is a Category B boule thrown contrary to the rules, which means that the offended team may choose whether or not to declare the boule to be dead, or simply to carry on with the game. As a practical matter, Team A will always declare a forgotten boule to be dead. There is no possible advantage for Team A in allowing Team B to throw its forgotten boule, and there is a genuine disadvantage— it would give Team B the boule advantage. For practical purposes, then, the rule of thumb is— a forgotten boule is dead and may not be played.
Note that the situation would be different if Team A had simply gone ahead and thrown its boule, without checking to see if Team B had any unplayed boules. In that case, Team A would have thrown out of turn. Team B would not have withheld a boule in violation of the rules, and it would have every right to go ahead and play its last boule. Whether Team B can declare Team A’s boule to be dead would depend on how you choose to treat a boule thrown out of turn.
The opinion expressed in this post is, I believe, the most reasonable interpretation of the FIPJP rules as they currently exist. Note, however, that there are a variety of opinions on this topic, and opinions may differ from player to player and umpire to umpire. In an umpired game, therefore, it is impossible to predict how the umpire will rule. For an opinion that Team A was at fault, because a team has the obligation to know who should play next before it plays, see the comments on THIS POST.