In the "Dictionary of Occupational Terms" produced by the UK Ministry of Labour, are listed Bath chair attendant, bath chair drawer, bath chair man, bath chair wheeler, chair drawer, chair wheeler; who pushed or drew an invalid chair or carriage. He could either own the chair ( Bath Chair Proprietor), hire one from a proprietor for a daily payment, or be employed by a proprietor at a weekly wage. They usually stood on a recognised rank with the chair for hire. There were also donkey bath chair drivers, who led a donkey drawing a bath, or invalid chair, and plied for hire chiefly in the streets of health resorts.
As MaryA has pointed out, you can use the 1881 Census on Ancestry to enter any search term you want in any one of the boxes.
Stan