Kirk, you wrote:
“All of our family records that were recorded in family Bibles and elsewhere specifically list his birth date as October 10th 1843 or 1845. Was it common for people in such a situation to be mistaken about their birthdays? “
Do you possess the actual artefacts (or scans of them) with the family information ? Have you ever looked at some of it with a degree of scepticism and wondered whether parts of the information were added non-contemporaneously, perhaps even by a person / people who weren’t alive when James Radcliffe b 1843 and his children were still alive ?
Have you studied each piece forensically, looking at handwriting styles to try and establish who wrote it and in which era ? If you haven’t already done this, it’s a good plan to take high resolution photos and scans of the writing and then view them on a screen---- for example, this would show up passages which were written in ballpoint pen.
You need to rule out the possibility that you could be relying on words which were written by family members who had hired a person on the IOM for the purpose of research into parish records (or visited the IOM, maybe back in the middle decades of the last century, to do the research themselves) which unearthed and fixed on the “James Radcliffe, son of John Radcliffe and Anne Corlett bap 10th October 1843 or 1845” and assumed that this was the right James------ who then proceeded to add the information into your family artefacts.
I’ve looked on Ancestry at the Manx part of the Kneale tree (with the 3rd cousin link) which you sketched out and there appear to be errors in it. Before you use it to compare gg and g g g grandparents with your own, you need to make sure that the Kneale tree information is absolutely correct. On your chart you wrote “3rd cousin 33cM / 3 seg”----what does that notation indicate ?