Better two sons for Katherine (aka Henry and Arthur/Ferdinand/Charles) then a Anne having a George/William/Thomas, Edward by Jane, John by Anna of Cleves (but he will marry her here? He is still Catholic and a king who has lost three wives in childbirth so Kristine of Denmark as fourth wife would be more likely than Anna of Cleves)... Katherine Howard can be a mistress and mother of the sixth son (or the sixth can be the second son of Kristine)
As I understand it Anne of Cleves was born Roman Catholic and converted to Lutheranism, even c. 1535, her mother was a devout Catholic whilst her brother was a practising Lutheran and Anne only formally converted to Anglicanism upon her arrival in England.
Catherine of Aragon dies (2), Anne Boleyn betrays (1), Jane Seymour dies in childbirth (1), Anne of Cleves survives (1), Catherine Howard is his mistress, and Catherine Parr never arrives on the scene, so perhaps ...
Catherine of Aragon:
1 - Henry IX, prev Prince of Wales, Duke of Cornwall
2 - Ferdinand, Duke of York
X - Mary Tudor, Princess Royal
Anne Boleyn:
3 - Thomas, Duke of Richmond & Somerset
X - Elizabeth Tudor
Jane Seymour:
4 - Edward, Bishop of Durham
Anne of Cleves:
5 - John, Duke of Ulster
Katherine Howard:
6 - Edmund Fitzroy, Constable of Beaumaris
I can't imagine a son of Anne of Cleves being sent to join the Church. Perhaps Anne Boleyn if she still does something that seriously upsets Henry VIII could cause the deligotimisation of Thomas, after all we've got Henry IX and Duke Ferdinand as Heir and Spare.
Would a 'bastard' such as Edmund Fitzroy or delegitimised issue such as Thomas be recognised with a title, or even able to rise up to the rank of Archbishop?
So let's say Thomas is made Duke of Richmond and gets stripped of his title whilst the unwed Elizabeth similarly finds herself unwed and unlikely to and forced to wait as a Lady in Waiting on Jane Seymour, Edmund is set up in a convenient out of the way position where he won't be the focus of much attention - Constable of Beaumaris, perhaps, like Henry VII rumoured illegitimate son, Roland De Velville, but he would be given this post by his half-brother Henry IX and therefore anything goes.
Edward therefore is our likely Bishop in the family. Perhaps my original idea of Bishop of Durham might be a better guess.