Another option might be to have number balance affect downtimes.
So 35 Bish, 25 Knit, 10 Rook:
Normal object (radar, ack, etc.) downtime is 30 minutes.
Bish hits Rook, downtime is 10/35 * 30 minutes = 8.6 minutes.
Bish hits Knit, downtime is 25/35 * 30 minutes = 21.4 minutes.
Rook hits Bish or Knit, downtime is 30 minutes (the normal time).
This would mitigate the number imbalance without depriving anyone of their favorite ride.
But, still wouldn't encourage swapping, or address LOCAL number imbalance or 2v1 ganging.