I have not been able to confirm a link between Capt. William Bowne and Anthony Bowne/Jane Radcliff,
Best I can tell, the child of Anthony Bowne (!st Count of Montague) and Jane Radclilffe named Anthony is: Anthony Browne (22 July 1552 – 29 June 1592) was Sheriff of Surrey and of Kent in 1580. He married Mary Dormer in 1572 and had the following issue: Anthony-Maria Browne, 2nd Viscount Montagu, John, Dorothy, Jane and Catherine. Also,the Anthony Bowne (1st Count of Montague) had a birthdate around 1528, seems odd to have had a child around 1570.
Sources:
G. E. Cokayne, ed. Geoffrey H. White, The Complete Peerage vol. XII, part I (London: St Catherine Press, 1953), p. 127; Richardson III 2011, p. 228.
Stopes, Charlotte Carmichael (1922). The Life of Henry, Third Earl of Southampton, Shakespeare's Patron. Cambridge: Cambridge University Press, p. 3.