You might want to leave it for a bit. BA last Tuesday announced a freight rate increase of 900% to start as of June 30th.
After some lobbying they relented and agreed only to raise their rates by 500%.
Cathay not an option as they are not accepting any pet shipments. Please note this is BA cargo local office here and not HQ in London.
If your pet shipment isn't urgent I would look at waiting it out for a bit.