We use a national petsitter (franchise) called Animals at Home who were suggested by the RWAF. Our local guy is really good, he came over before we went on holiday and brought all his various certificates (training/insurance), he also brought a notebook and took notes on the buns while we were chatting (feeding regime, habits, behaviours, cleaning out requirements, etc). I wouldn't say he was necessarily 'rabbit savvy', but our girlie bun had not long had surgery and had only just finished her meds. He was honest and said that he had no experience medicating buns and would not be comfortable medicating a bun (he does cats though which surely has to be harder!!), however he is also the pet ambulance for our own vet so we told him what to look out for in the event that either of them were 'under the weather', and he would take them to the vet and if required the vet could show him how to administer. Possibly others would be happier to medicate buns with your guidance.
Looking at the website it looks like the only have people in east and south Birmingham but maybe worth contacting them in case its not up to date.