Dog to the UK
Hi Billy,
You unfortunately have two requests which do not go together i.e. Flying an animal and cheap as possible.
The only carriers worth considering are BA and VS in that order. BA has far more experience than any other airline and the reputation of looking after AVI properly. Both BA & VS are direct flights and consequently much fairer on the animal which should be the major concern here. I presume when you say you have the paperwork sorted you mean the animal is chipped and entering under the Pets Passport scheme to prevent 6 months quarantine.
I shouldn't worry too much about flying them in the belly holds, i think the holds drop 2deg per sector hour from the ambient temp at departure airport which given Vegas, will be hot anyway but never known a crew to forget the hold heating when animals are being carried. Far more likely to bake or freeze the great unwashed upstairs.
Regarding costs, all airlines normally charge the same which is based on the IATA rate plus surcharge and the kennel size has to conform to the AVI regulations which if I remember right, must be:-
Length = tip of animals nose to base of tail
Width = distance across animals shoulder x 2
Height = bottom of foot to tip of ears plus half again.
The weight is not really important as the airlines will charge you based on volume I.e. H x W x L (in cms) divided by 6000 gives you the volume weight in kilos. Count on around $20 per kilo plus charges.
The above may be out of date a bit so check the rules with the airlines or DEFRA
Hope this helps & Good Luck