A qualified electrician should be doing this job. Only he will have the equipment to test the resistance of the earthing connection. Such a person ought not to be confused.
Personally I don't see why they are talking about a new earthing rod outside - if the resistance to earth was low enough before, I don't see why they are needing to break the previous earth connection - unless you have had a new consumer unit (meter) installed as well.
Neither do I see why they are seeking to charge you extra for this. If they quoted for the job, they quoted for the whole job - unless there was some caveat pointing this out on the Order Confirmation?