Transporting pets US to UK (Cambridge)

I am reading nightmare advice on other message boards about pets costing $3-4,000 per animal because most Airlines require you to use a Pet Relocation service. Really?? I know my furry friend needs Rabies and worming and microchip. I have not looked into flying my pet yet. I just started looking at our flights. Kuwait offers $750 one way with few days stopover in NYC then on to England. But Kuwait Airlines not on approved list for transporting pets. Has anyone done it recently without relocation service?