Hi David
And I didn't pick up that you'd said brother, not son, either. Do you have the details of William's being made a burgess, because that's information I haven't been able to unearth so far, and it would be really helpful? I'm also intrigued by your mention of Arthur being married to a Jane Badger. The info I have, to date, is that he married Sarah Wakely in Honiton in 1753 (died 1777, Honiton) and then Elizabeth Morgan at All Saints, Bristol, by licence in 1781. Elizabeth is the "dear wife" mentioned in his will.
Arthur's brother William was a peruke maker and hair dresser and I've established that he was in business in Bath between 1760 or so, and 1769, possibly later, though I can't find any advertisements or mention of him in Bath beyond that year. Where he did his apprenticeship, I do not know. I do know, however, that he moved to Bristol at some point before he died in 1791, but not when exactly. This William's son John was a tailor and stay maker in Bristol prior to 1796 when he and his family moved to America; I've found very little on John's life in England, though the American side is very well documented through his work as a nonconformist preacher, and as a collector of funds (and books) for establishing Miami University. Yesterday I found a trade card for Wall and Browne, "Stay Makers, Taylors and Habit Makers," of Broad Mead, Bristol, so this morning I'm hoping to follow up the Walls, to try and identify who his partner was.