You cannot do what you propose (ship pet as "baggage" on UA and then as cargo on BA) unless your pet enters the UK (clears customs and quarantine, not sure what the current rules are but used to be 6 months). To "interline" you will have to pay for cargo from IAD to EVN (likely to be triple the "baggage" price). There is one fee (to UA) and one set of paperwork, and it is taken care of by UA. They will charge you a fee for the transfer from UA to BA at LHR. No significance to being a BMed flight, it is BA as far as you are concerned. This will be expensive.