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My husband used to work for a company that managed cemeteries (in England).  Occasionally interred remains had to be removed, and permissions were required. A licence from the bishop (I imagine this was for Church of England cemeteries) cost £1000 and disinterments have to be done in the very early hours of the morning. Even if someone has been buried for 100 years or so, it is a very unpleasant task - the smell is dreadful - so the four or five gravediggers are paid handsomely. Several years ago, it cost upwards of £2000.
Then as others have said, you would need specialists to open the coffin and remove appropriate material for DNA analysis - I've no idea what the cost of that would be, but you must be talking hundreds of pounds.
But I think your biggest stumbling block would be that permission would not be granted because "the authorities" would consider your request not to be justified, not to mention all the ethical considerations.
I read and understand that you feel you have no reliable facts to start a paperchase, but Rootschatters on this forum have tracked down, with a high degree of certainty, other individuals who did not leave behind a standard, reliable paper trail. I think that is certainly a route worth exploring.....

Generally it looks like correspondence is kept for 10 years https://www.gov.uk/government/publications/10-downing-street-correspondence-communications-gifts-and-hospitality-privacy-notice/10-downing-street-correspondence-communications-gifts-and-hospitality-privacy-notice

But I wonder if he also sent a copy of the poem to a local newspaper or similar?

And if he wrote one poem, there may have been more?  Is there anything in local archives, literary society archives etc?

Hi
I had a look for burials on Ancestry
Kenneth Mc Kenzie buried Deptford St Nicholas  of King St 24 Oct 1837 (there is death cert Greenwich Dec qtr 1837  5 159 which may help)
There is unfortunately no age

quote]

This Kenneth Mackenzie was 76 at the time of his death (see GRO index)

Apologies if the following  sounds a bit tactless but....your Dad would not yet be 100 years old, and so his name will be redacted (ie his name will not be visible) unless the managers of the register know with certainty that he is deceased.