I don't know if this will help, but I do have a transport record for a John Byrom in 1655:
"Cavaliers and Pioneers" Patent Book No. 3, p.318 Mr. Thomas Purifoy, 2,000 acs. on S. side of Rappa. Riv., in the freshes, about 4 mi. above Nanzemum Towne., 13 Mar 1655, p. 386. Trans. of 40 pers: William Elliott, George Pennell, Ann Freeman, Ann Tawney, John Byrom, Joseph Lansey, 3 times, Ann Bowles, Senr., Ann Bowles, Junr., Charles Aston, Issabel his wife, John his son, Elinor Steward, 4 Negroes; Wm. Underfield, Mary Slaughter, Edmond Sweny, Senr., Eliz. his wife, Edmond Sweny, Junr., Elizabeth Sweney, Mary Sweney, Jane Richards, Augustine Johannes Van de Powke.