The link posted was from this web site
https://www.gov.uk/take-pet-abroad
On the documented list you are viewing the right hand column is the important one. This lists the airports which are authorised for import under the PETS scheme. See which airport you would like to enter via then look into the left column to see which company you would like to use.
Basically as far as the UK are concerned you can leave from any airport overseas but must enter the UK via an approved airport where they check your pet's microchip and passport. You can land at the entry airport then take off again and fly to another destination in the UK. So for example fly from the south of France in to Biggin Hill then fly onwards to another airport in the UK.
It is perfectly achievable and legal. It is even easy if you know what you are doing. If any charter operators are not being helpful it is probably because they do not know what they are doing or are not approved under the PETS scheme. Just put the phone down and move on to your next contact.
I am afraid I am out of the business now so can not offer any personal recommendation.
MM