John Rainolds was the one who purposed the new translation as a unifying Bible at the Hampton Court Conference. About one fourth of the translators chosen were favorable towards the Puritians.I don't think you have this quite right. At the Hampton Court conference in 1604, James declared to the Puritan party, "If this be all your party have to say for themselves, I will make them conform to the Church, or I will harry them out of the land, or else do worse." James was a calvinist, but he hated Presbyterianism because of his problems with Scottish Presbyterianism. Therefore he wanted an Episcopal settlement for the English Church (the status quo) so that he could keep the Church under his thumb.
FYI, 'English' is England only; 'British' and 'Britain' refers to England, Scotland, Wales and Northern Ireland. Therefore, since I live in England, I am both English and British. Until the Act of Union between England and Scotland in 1707, there was no 'Britain,' though people did refer to 'the British Isles.'
I do not mean that King James viewed Puritians and Presbyterians favorably. He did not hold the detaine Barclay held (Barclay did not want a new translation and abhored Calvinists).
I mean King James sought to unify factions under one Bible.