Flying with a cat in cabin to the United Kingdom?

I'm moving to the UK in about 5 months and I'm starting to learn that bringing my pets (Dog and cat) with me will be an extraordinary challenge that I did not expect.

I'm more concerned for my cat who will need to fly in cabin with us due to his anxiety. I've learned that the UK has explicitly banned the travel of all pets in the cabin (including those deemed as 'emotional support'). From my understanding our only options are to:

1. Fly our cat in the hold (which actually isn't really much of an option for us)

2. Fly to Paris, drive to the Eurotunnel, take the Eurotunnel Le Shuttle, continue driving to London. Total travel from Paris>London 6 hours.

Has anyone here had any experience moving with a cat from the US to the UK? Are there any other options I'm not thinking of?

Appreciate any help.