I've had a look through these newspaper articles, which are fascinating. So John Cater junior either died suspiciously 'at home' and was buried near his home, or died in the army in the West Indies. I tend towards the latter. It warrants a look at some West Indies ancestry sites and fh sources.
As curious as those stories are about the skeleton, I'd tend to agree with you, that the family most likely knew what really happened to John.
If you have access to the Kempston records, cuffie, I wonder if you would be prepared to have a look for any 17th c. Kendall baptisms and burials in that parish.
There's no Kendall baptisms and only one burial, which likely isn't related to your Kendalls.
Burials - All Saints, Kempston:
09 Nov 1793 Susan Kendal; dau. of Thomas (pauper) & Ann
As an aside, there's no Beaumont baptisms and only one burial.
Burials - All Saints, Kempston:
11 Feb 1690 Mary Beaumont; widow of Benjamin
Margaret had a great aunt, also called Margaret Beaumont who married a John Kendall of Kempston. She was born about 1644 but I have no other details. according to a few online refs. They apparently had 3 daughters. I can find no other reference to them on Ancestry or by Googling them.
There was a Samuel Cater of Kempston Hall and Harold, Sheriff of co. Beds., who was bap. Kempston 1/3/1645, died 1704, who married an Anne Cater, daughter John Kendall of Bassingbourn, co. Cambridge. I wondered if this Anne was one of the 3 daughters of John Kendall and Margaret Beaumont??
Would you be able to have a look at the Kempston records between 1650 and 1700 to see if there is any reference to this couple and their daughters? I understand if you'd rather not.
As mentioned, there's no Kendall baptisms at Kempston and only that one burial. There's a few marriages but none seem to have an obvious connection to the family you're looking for.
Marriages - All Saints, Kempston:
02 Jul 1571 Jn Kendall and Ellen Carter
06 Sep 1624 Chris Smyth and Kath Kendall
14 May 1781 Tho Kendall and Ann Radley (widower)
As mentioned on post #2, the best option for finding Beds. baptisms, marriages and burials is to start with the Beds. FHS' surname index, as it helps to quickly identify any possible records. Then use that to find the details in transcripts or scans (I don't think there's too many PR scans online though).