You can't remove a label with a filter.

However, it seems that Gmail applies filters sequentially, so if you change the order of your filters so that...

• the @BitBucket filter is listed above the @Private filter, and
• the @Private filter includes a negative search in its search criteria (e.g., -label:@BitBucket) so that it only applies the label to messages that don't already have the @BitBucket label

I think that will do what you need.

As for changing the order of your filters, see: How can I reorder Gmail filters?

You can't remove a label with a filter.

However, it seems that Gmail applies filters sequentially, so if you change the order of your filters so that...

• the @BitBucket filter is listed above the @Private filter, and
• the @Private filter includes a negative search in its search criteria (e.g., -label:@BitBucket) so that it only applies the label to messages that don't already have the @BitBucket label

I think that will do what you need.

As for changing the order of your filters, see: How can I reorder Gmail filters?