Currently Bitbucket sends batched notifications based on few parameters:
plugin.bitbucket-notification.batch.min.wait.minutes

The minimum time to wait for new notifications in a batch before sending it (inactivity timeout). The default is 10 minutes.

plugin.bitbucket-notification.batch.max.wait.minutes

The maximum time to wait since the first notification of a batch before sending it (staleness avoidance timeout). The default is 30 minutes.

plugin.bitbucket-notification.batch.notification.flush.limit

The maximum number of notifications collected in a batch before the batch is sent automatically. Once this number of notifications is reached, the batch will be sent regardless of the time settings. The default is 40.

But in case the mail.max.message.size is reached, the notification is dropped and the user does not receive any kind of error, there is just an INFO message logged in atlassian-bitbucket.log

INFO  [Caesium]  c.a.s.i.n.r.b.RepositoryBatchSender REPONAME: Dropping 40 notification(s) for USERNAME; the message is too large (1699116 > 1048576)

It would be good if Bitbucket can handle those exceptions and send a small note letting the users know the complete notification was not sent due to its size. They would be able to manually check the repository activity if necessary.