You will need a vet invoice or receipt from whom ever you fly with. You Can't just impose a rate on what you paid as that amount you pay includes money for tax and ins etc.
As flights are exempt from VAT i.e no VAT was actually charged to you then you are not in a position to claim back the VAT
as per howareya you cannot impose a VAT charge you must actually have been charged VAT and have an invoice supporting same