You have choices other than BA & Virgin if you fly from LA to another US city first. If you are worried about the long flight, this might be a good option. Also, with the rules regarding deworming prior to your animal entering the UK, a long flight may run the risk of delays and cancellations, and your animal would have to be dewormed again before being released into your custody on the other side of the pond.
If you fly from LA to Boston, for example, you can have your dog dewormed in Boston, stay overnight, and fly out the next day. IcelandAir is an approved animal carrier for the Boston to London route, and they have a short layover in Iceland, which may ease your worries about the dog watering issue.
Also one plus about using IcelandAir is that your animal can be checked as extra baggage, and the fees/cost of your animals transportation would decrease significantly.