There's another page in the Peerage that lists all of these Jameses in order one right after the other, but I can't find it at the moment so you'll need to click on James II below him and then James III on the subsequent page.
I might be misinterpreting this, and of course there are many other sources I have yet to explore.
Here's the page with the info. on Janet Bethune (Beaton):
I just noticed that James I did have an illegitimate daughter named Margaret Hamilton from his relationship with Beatrix Drummond (I must have missed that page last time), but once again the birth year is not compatible (1505).
Looking forward to hearing your thoughts on the Wier post too.